PRACTICAL  ASTRONOMY 


BY 


P.  S.  MICHIE       AND       F.   S.   HARLOW 

Late  Professor  of  Ph  ilosofhy  1st  L  ieut.  1st  A  rtillery  U.  S.  A. 

U.  S.  M.  A. 


THIRD  EDITION  REVISED 

THIRD    THOUSAND 


NEW  YORK 

JOHN  WILEY  &  SONS,  INC. 
LONDON:  CHAPMAN  &  HALL,  LIMITED 


COPYRIGHT,  1893, 

BY 

P.    S.   MICHIE   AND  F.    S.    HARLOW. 


•  *  :  s  **•  • 


PRESS    OF 

BRAUNWORTH   &   CO. 

BOOKBINDERS    AND    PRINTERS 

BROOKLYN.    N.    Y. 


PREFACE. 


THIS  volume,  both  in  respect  to  matter  and  arrangement,  is 
designed  especially  for  the  use  of  the  cadets  of  the  U.  S.  Military 
Academy,  as  a  supplement  to  the  course  in  General  Astronomy  at 
present  taught  them  from  the  text-book  of  Professor  0.  A.  Young. 
It  is  therefore  limited  to  that  branch  of  Practical  Astronomy  which 
relates  to  Field  Work,  and  more  particularly  to  those  subjects  which 
are  not  discussed  at  sufficient  length  for  practical  work  in  Professor 
Young's  volume.  It  is  believed,  however,  that  it  will  find  a  use- 
ful application  in  the  hands  of  officers  of  the  Army,  who  may  be 
called  upon  to  conduct  such  explorations  and  surveys  for  military 
purposes  as  the  War  Department  may  from  time  to  time  direct. 

The  more  usual  methods  of  determining  Time,  Latitude,  and 
Longitude,  on  Land,  are  explained,  and  the  requisite  reduction 
formulas  are  deduced  and  explained.  In  addition,  there  is  given  a 
short  explanation  of  the  principles  relating  to  the  Construction  of 
Ephemerides,  to  the  Figure  of  the  Earth,  the  determination  of 
Azimuths,  and  the  projection  of  Solar  Eclipses. 

The  instruments  described  are  those  used  by  the  cadets  in  the 
Field  and  Permanent  Observatories  of  the  Military  Academy  dur- 
ing the  summer  encampment. 

The  principal  sources  of  information  from  which  the  matter  in 
this  volume  has  been  derived  are  the  published  Reports  of  the 
United  States  Lake,  Coast,  and  Northern  Boundary  Surveys;  the 
publications  of  the  Hydrographic  Office,  U.  S.  Navy,  and  the 
works  of  Brlinnow  and  Chauvenet. 

U.  S.  MILITARY  ACADEMY, 
WEST  POINT,  N.  Y.,  October,  1892. 

lit 


CONTENTS. 


EPHEMERIS. 

PAOE 

American  Ephemeris  and  Nautical  Almanac „ ,  1 

Ephemeris  of  the  Sun. 

Deduction  of  Formulas  for  Computation  of  the  Sun's  Tables. 2 

Table  of  Epochs 5 

Table  of  Longitude  of  Perigee 5 

Table  of  Equations  of  the  Center 5 

Perturbations  in  Longitude  ;  Aberration 6 

Ephemeris  of  the  Sun 6 

Earth's  Radius  Vector 6 

Sun's  Horizontal  Parallax 7 

Sim's  Apparent  Semi-diameter , 7 

Equation  of  the  Equinoxes  in  Longitude 7 

Equation  of  Time 9 

Ephemeris  of  the  Moon. 

Elements  of  the  Lunar  Orbit 10 

Ephemeris  of  the  Moon 12 

Ephemeris  of  a  Planet.  13 

INTERPOLATION.  15 

THE  TRANSIT. 

Description  of  the  Transit  Instrument 17 

The  Reticle 19 

The  Eye-piece  and  Setting  Circles , 20 

Adjustments  of  the  Transit. 

1.  To  Place  the  Wires  in  the  Principal  Focus  of  the  Objective 21 

2.  To  Level  the  Axis 21 

3.  To  Place  the  Wires  at  Right  Angles  to  the  Rotation  Axis 23 

v 


VI  CONTENTS. 

PAGE 

4.  To  Place  the  Middle  Wire  ID  the  Line  of  Collimation 23 

5.  To  Place  the  Line  of  Collimation  in  the  Meridian 23 

INSTRUMENTAL  CONSTANTS. 

1.  The  Value  of  One  Division  of  the  R.  A.  Micrometer  Head 24 

2.  The  Equatorial  Intervals. . '. -** 27 

3.  The  Reduction  to  the  Middle  Wire. . .  1 28 

4.  The  Value  of  One  Division  of  the  Level 29 

5.  Inequality  of  the  Pivots 30 

EQUATION  OF  THE  TRANSIT  IN  THE  MERIDIAN. 

1.  The  Effect  of  an  Error  in  Azimuth  on  the  Time  of  Passage  of  the 

Middle  Wire 35 

2.  The  Effect  of  an  Inclination  of  the  Axis  on  the  Time  of  Passage  of 

the  Middle  Wire 34 

3.  The  Effect  of  an  Error  in  Collimatiou  on  the  Time  of  Passage  of  the 

Middle  Wire 34 

Determination  of  Instrumental  Errors. 

1.  To  Determine  the  Level  Error 37 

2.  To  Determine  the  Collimation  Error 37 

3.  To  Determine  the  Azimuth  Error 39 

REFRACTION  TABLES 40 

TIME. 

Relation  between  Sidereal  and  Mean  Solar  Intervals 43 

Relation  between  Sidereal  and  Mean  Solar  Time 44 

Example  Solved 46 

To  FIND  THE  TIME  BY  ASTRONOMICAL  OBSERVATIONS. 

/.  Time  by  Meridian  Transits. 
1st  Method.  To  Find  the  Error  of  a  Sidereal  Time-piece  by  the  Meridian 

Transit  of  a  Star  (Form  1,  Appendix) 47 

•fr  To  Find  the  Same  by  the  Meridian  Transit  of  the  Sun 48 

2d  Method.  To  Find  the  Error  of  a  Mean  Solar  Time-piece  by  a  Merid- 
ian Transit  of  the  Sun  (Form  2) 49 

*  To  Find  the  Same  by  a  Meridian  Transit  of  a  Star 50 

THE  SEXTANT. 

Description  of  the  Sextant 50 

How  to  Measure  an  Angle  with  the  Sextant 54 

Adjustments  of  the  Sextant. 

1.  To  Make  the  Index-glass  Perpendicular  to  the  Frame 56 

2.  To  Make  the  Horizon-glass  Perpendicular  to  the  Frame 56 


CONTENTS. 


3.  To  Make  the  Axis  of  the  Telescope  Parallel  to  the  Frame 56 

4.  To  Make  the  Mirrors  Parallel,  when  the  Reading  is  Zero 57 

Errors  of  the  Sextant. 

Index  Error . ,  57 

Eccentricity 58 

The  Astronomical  Triangle , 58 

II.   Time  by  Single  Altitudes. 

1st  Method.  To  Find  the  Error  of  a  Sidereal  Time-piece  by  a  Single  Alti- 
tude of  a  Star  (Form  3) 59 

•f-  The  Correction  to  be  Applied  to  the  Mean  of  the  Altitudes. .  60 

To  Ascertain  what  Stars  are  Suitable  for  this  Method 63 

2d  Method.  To  Find  the  Error  of  a  Mean  Solar  Time-piece  by  a  Single 

Altitude  of  the  Sun's  Limb  (Form  4) 64 

III.  Time  by  Equal  Altitudes. 

1st  Method.  To  Find  the  Error  of  a  Sidereal  Time-piece  by  Equal  Alti- 
tudes of  a  Star  (Form  5) 66 

3d  Method.  To  Find  the  Error  of  a  Mean  Solar  Time-piece  by  Equal  Alti- 
tudes of  the  Sun's  Limb  (Form  6) 67 

•J*  Correction  for  Refraction 68 

•fr  Equation  of  Equal  Altitudes 69 

Time  of  Sunrise  or  Sunset 70 

Duration  of  Twilight 70 

LATITUDE. 

Form  and  Dimensions  of  the  Earth 72 

The  Eccentricity  of  the  Meridian 74 

The  Equatorial  and  Polar  Radii 74 

The  Radius  of  Curvature  of  the  Meridian  at  the  Observer's  Station 75 

The  Length  of  a  Degree  of  Latitude  75 

The  Length  of  a  Degree  Perpendicular  to  the  Meridian 75 

The  Length  of  a  Degree  of  Longitude 76 

The  Length  of  the  Earth's  Radius  at  any  Point 76 

The  Reduction  of  Latitude 77 

Latitude  Problems. 

1st  Method.  By  Circumpolars 78 

2d  Method.  By  Meridian  Altitudes  or  Zenith  Distances 78 

3d  Method.  By  Circum-meridian  Altitudes 79 

Formula  for  Reduction  to  the  Meridian 79 

Method  of  Making  and  Reducing  Observations 80 

Hour  Angle  and  Correction  for  Clock  Rate 81 

By  Circum-meridian  Altitudes  of  the  Sun's  Limb  (Form  7). .  83 

By  Circum-meridian  Altitudes  of  a  Star  (Form  8) 84 

i 


Till  CONTENTS. 

PAGE 

Notes  on  this  Method. 

1.  Ephemeris  Star  Preferable  to  Sun 84 

2.  Advantage  of  Combining  the  Results  from  Two  Stars 84 

3.  Advantage  of  Selecting  Stars  Distant  from  Zenith 84 

4.  Reduction  of  Mean  Solar  to  Sidereal  Intervals 86 

5.  To  Determine  the  Reduction  to  the  Meridian 87 

THE  ZENITH  TELESCOPE. 

Description  of  the  Zenith  Telescope 90 

The  Attached  Level  and  Declination  Micrometer 91 

4th  Method.  By  Opposite  and  Nearly  Equal  Meridian  Zenith  Distances. 

Captain  Talcott's  Method  (Form  9) 96 

Conditions  for  Selecting  a  Pair  of  Stars 97 

Preliminary  Computations 97 

Adjustment  of  Zenith  Telescope 98 

Observations 99 

Reduction  of  Observations 99 

1.  Reduction  from   Mean   Declination  to   Apparent   Declination  of 

the  Date 99 

2.  The  Micrometer  and  Level  Corrections 100 

3.  The  Refraction  Correction 102 

4.  The  Correction  for  Observations  off  the  Meridian 103 

•%•  To  Determine  the  Reduction  for  an  Instrument  in  the  Meridian. . .  105 

*  To  Determine  the  Probable  Error  of  the  Final  Result 106 

5th  Method.  By  Polaris  off  the  Meridian  (Form  10) 109 

6th  Method.  By  Equal  Altitudes  of  Two  Stars  (Form  11) 114 

LONGITUDE. 

1st  Method.  By  Portable  Chronometers 118 

2d  Method.  By  the  Electric  Telegraph  (Form  12) 121 

Reduction  of  the  Time  Observations  (Form  12a) 125 

•%•  Personal  Equation 127 

4*  Application  of  Weights  and  Probable  Error  of  Result 128 

3d  Method.  By  Lunar  Culminations 132 

Observations  and  Reductions 135 

Equation  of  Transit  Instrument  Applicable  to  this  Method...  137 

4th  Method.  By  Lunar  Distances 140 

1.  Correction  for  Moon's  Augmented  Semi-diameter 141 

2.  Correction  for  Refraction ,...., 142 

3.  Correction  for  Earth's  Oblateness 142 

Explanation  of  this  Method 142 

Observations , 148 

•r  To  Find  Augmentation  of  Moon's  Semi-diameter 149 

•%•  To  Deduce  the  Law  of  Refractive  Distortion 150 

•*•  To  Deduce  the  Parallax  for  the  Point  R. 151 

•J-  To  Determine  the  Correction  for  Earth's  Oblateness 151 


CONTENTS.  ix 

OTHER  METHODS  OP  DETERMINING  LONGITUDE. 

PAGE 

1.  By  Signals 153 

2.  By  Eclipses  and  Occultatious 153 

3.  By  Jupiter's  Satellites 153 

a.  From  their  Eclipses 153 

b.  From  their  Occupations 153 

_*     c.  From  their  Transits  over  Jupiter's  Disc 154 

d.  From  the  Transit  of  their  Shadows 154 

Application  to  Explorations  and  Surveys •. 154 

TIME  OP  OPPOSITION  OR  CONJUNCTION.  156 

TIME  OF  MERIDIAN  PASSAGE.  .  157 

AZIMUTH. 

Definitions 158 

The  Astronomical  Theodolite  or  Altazimuth , 159 

Classification  of  Azimuths 160 

Selection  of  Stars 160 

Measurement  of  Angles  with  Altazimuth 1 62 

Observations  and  Preliminary  Computations 164 

REDUCTION  OP  OBSERVATIONS.  165 

*  1.  Diurnal  Aberration  in  Azimuth 166 

2.  To  Reduce  an  Azimuth  observed  shortly  before  or  after  the  Time 

of  Elongation,  to  its  Value  at  Elongation 167 

DECLINATION  OP  THE  MAGNETIC  NEEDLE.  168 

SUN-DIALS.  168 

Values  of  Equation  of  Time  to  be  added  to  Sun-Dial  Time 173 

SOLAR  ECLIPSE. 

Solar  Ecliptic  Limits 174 

PROJECTION  OP  A  SOLAR  ECLIPSE. 

1.  To  find  the  Radius  of  the  Shadow  on  any  Plane  Perpendicular  to  the 

Axis  of  the  Shadow 176 

2.  To  find  the  Distance  of  the  Observer  at  a  given  time  from  the  Axis  of 

the  Shadow 178 

3.  To  find  the  Time  of  Beginning  or  Ending  of  the  Eclipse  at  the  Place 

of  Observation 180 

4.  The  Position  Angle  of  the  Point  of  Contact 182 

5.  The  necessary  Equations  for  Computation  arranged  in  order  for  the 

Solution  of  the  Problem 182 

TABLES.  185 

FORMS.  203 


PRACTICAL  ASTBONOMY. 


EPHEMERIS. 

Ephemeris. — The  numerical  values  of  the  coordinates  of  the 
principal  celestial  bodies,  together  with  the  elements  of  position  of 
the  circles  of  reference,  are  recorded  for  given  equidistant  instants 
of  time  in  an  Astronomical  Ephemeris. 

The  "American  Ephemeris  and  Nautical  Almanac"  is  pub- 
lished by  the  United  States  Government,  generally  three  years  in 
advance  of  the  year  of  its  title,  and  comprises  three  parts,  viz. : 

Part  I. — Ephemeris  for  the  Meridian  of  Greenwich,  which  gives 
the  heliocentric  and  geocentric  positions  of  the  major  planets,  the 
ephemeris  of  the  sun,  and  other  fundamental  astronomical  data  for 
equidistant  intervals  of  mean  Greenwich  time. 

Part  II. — Ephemeris  for  the  Meridian  of  Washington,  which 
gives  the  ephemerides  of  certain  fixed  stars,  sun,  moon,  and  major 
planets,  for  transit  over  the  meridian  of  Washington,  and  also  the 
mean  places  of  the  fixed  stars,  with  the  data  for  their  reduction. 

Part  III. — Phenomena,  which  contains  prediction  of  phenomena 
to  be  observed,  with  data  for  their  computation. 

'EPHEMERIS  OF  THE   SUN. 

To  construct  the  ephemeris  of  the  sun  it  is  necessary  to  com- 
pute its  tables:  these  are 

1.  The  table  of  Epochs. 

2.  The  table  of  Longitudes  of  Perigee. 

3.  The  table  of  Equations  of  the  Center,  and  its  corrections. 

4.  The  table  of  the  Equations  of  the  Equinoxes  in  Longitude. 


2  PRACTICAL  ASTRONOMY. 

In  Mechanics*  it  was  shown  that  the  Earth's  undisturbed  orbit 
is  an  ellipse,  having  one  of  its  foci  at  the  sun's  center,  and  that  the 
earth's  angular  velocity  is 


its  radius  vector, 


+  e  cos  6' 
its  constant  double  sectoral  area, 

li  —   |V  «(1  —#*)', (615) 

and  its  periodic  time, 

//y""        2  TT 

r  =  2  n\/-,  =  *-? (616) 

In  these  expressions  0'  is  the  angle  made  by  the  earth's  radius 
vector  with  any  assumed  right  line  drawn  through  the  sun's  center, 
6  that  included  between  the  radius  vector  and  the  line  of  apsides 
estimated  from  perihelion,  and  n  is  the  mean  motion  of  the  earth 
in  its  orbit. 

From  (551),  (615)  and  (616),  we  have 


-f  e  cos 


dt  ~  dt  ~  a*  (1  -  e2)2 


/ 
V 


//?  (1  +  e  cos  BY  _      (1  -f  e  cos  By 
(!-«•)•  (1  -  *')» 


and  therefore 

n  d  t  =  (1  -  e2)*  (1  +  e  cos  6y*  d  0'.  (2) 

Since  e  varies  but  little  from  0.01678  (see  Art.  185,  Young  f),  we 
may  omit  all  terms  containing  the  third  and  higher  powers  of  e  in 
the  development  of  the  second  member  of  the  preceding  equation. 

*  Micliie's  ^Mechanics,  4th  Edition. 
,f  Young's  General  Astronomy. 


SPHEMERI8. 


3 


Then  after  substituting 


2 


for  cos2  9,  we  have 


n  dt  =  d  d'  -  2  e  cos  Bd  9  -f  |ea  cos  2  Bd  (2  0)  +  etc.         (3) 
Integrating  we  have 


n  t  +  0  =  d'  -  2e  sin  6  +  f  e2  sin  2  8  +  etc. 


(4) 


The  earth's  orbit  is,  however,  not  entirely  undisturbed.  Due  to 
the  perturbating  action  of  other  bodies  of  the  solar  system  the  earth 
is  never  exactly  in  the  place  which  it  would  occupy  in  an  undis- 
turbed orbit.  Moreover  the  line  of  apsides  has  a  direct  motion,  i.e., 
in  the  direction  in  which  longitudes  are  measured,  of  about  11".7 
per  annum,  and  the  vernal  equinox  an  irregular  retrograde  motion 
whose  mean  value  is  about  50".2  per  annum. 

Therefore  (Fig.  1),  let  the  line  from  which  0'  is  estimated  be 


that  drawn  through  the  sun  and  the  position  of  the  mean  vernal 
equinox  V  at  some  fixed  instant,  called  the  epoch.  Then  when 
6  is  zero,  0'  will  be  the  longitude  of  perihelion,  estimated  from  this 
point.  Let  this  be  denoted  by  lp ,  and  the  time  of  perihelion  pas- 
sage by  tp\  then  from  (4)  we  have, 


nt9+C=lp. 


(5) 


4  PRACTICAL  ASTRONOMY. 

Subtracting  from  (4)  we  have 

n  (t  -  g  =  d'  -  lp  -  2  e  sin  0  +  f  ea  sin  2  0,  (6) 

which  since 

v-ip=e  17) 

reduces  to 

n  (*  -  y  =  (0'  -  lp)  -  2  esin  (0'  -  lp)  +  ie'sin2  (P  -  lp).     (8) 
Transposing  lp  ,  we  have 
71  (Z  -  g  +lp  =  lm=e'-Ze*m  (0'  -  lp)  +  f  e'sin  2  (0'  -  *,),    (9) 

in  which  /w  is  the  longitude  of  the  mea/n  place  of  the  earth  at  the 
time  I,  referred  to  the  same  origin. 

Let  L  be  the  longitude  of  the  earth's  mean  place  at  the  epoch, 
also  referred  to  the  same  origin,  and  T  any  interval  of  time  before 
or  after  this  epoch.  Then  will 

lm  =  L+nT,  (10) 

and  we  have 

L  +  nT  =  &'  -2esm  (6'  -  lp)  +  f  e2  sin  2  (0'  -  lp).         (11) 

To  find  the  values  of  the  four  unknown  quantities,  L,  n,  e,  and 
lp  ,  take  four  observations  of  R.  A.  and  declination  at  different  times, 
and  having  reduced  the  declination  to  its  geocentric  value  by  cor- 
recting for  refraction  and  parallax,  find  the  corresponding  longi- 
tudes (Art.  180,  Young). 

Each  longitude  is  necessarily  referred  to  the  true  equinox  of  its 
own  date.  Eeduce  each  to  the  mean  equinox  of  the  epoch  by  cor- 
recting for  aberration,  nutation,  precession,  and  perturbations,  add 
180°,  and  the  results  will  be  the  longitudes  of  the  true  place  of  the 
earth  referred  to  a  common  point—  the  mean  equinox  of  the  epoch. 

They  will  therefore  be  the  values  of  0'  corresponding  to  the 
values  of  T  in  the  following  equations,  the  solution  of  which  will 
give  L,  n,  e,  and  lp. 

L      n  7    =  #-2esin    0    - 


L  +  n  T,  =  6J  -  2  e  sin  (0/  -  lp)    ,  .     . 

L  +  nT,=  0,'  -  2  esin  (0/  -         '  * 


BPIIEMERIS.  5 

The  value  of  n  derived  from  these  equations  is  evidently  the 
earth's  mean  motion  from  a  fixed  point. 

Its  mean  motion  from  the  moving  mean  vernal  equinox  (or 
mean  motion  in  longitude)  is  evidently  given  by 

360° 


"  360°  -  50."2" 

These  observations  repeated  at  different  times  will  determine 
the  changes  that  take  place  in  w,  e,  and  lp\  from  the  last  two  the 
variations  in  the  eccentricity  and  the  rate  of  motion  of  perihelion 
can  be  found. 

Having  in  this  manner  found  the  elements  of  the  earth's  place 
and  motion,  the  corresponding  mean  longitude  of  the  sun  at  any 
instant  can  be  obtained  by  adding  to  that  of  the  earth  180°. 
L  +  n'  T-\-  180°  will  then  give  for  any  instant  the  mean  longi- 
tude of  the  sun's  mean  place.  The  difference  between  the  longi- 
tudes of  the  sun's  true  and  mean  places  at  any  instant  is  the 
Equation  of  the  Center  for  that  instant. 

From  the  preceding  elements  let  it  be  required  to  construct  the 
Tables  of  the  Sun. 

1.  The  Table  of  Epochs. — Take  mean  midnight,  December  31 — 
January  1,  1890,  as  the  epoch.     To  the  mean  longitude  of  the  sun's 
mean  place  at  that  epoch,  add  the  product  of  the  sun's  mean  motion 
n',  by  the  number  of  mean  solar  days  after  the  epoch,  subtracting 
360°  when  this  sum  is  greater  than  360°.     These  longitudes  with 
their  corresponding  times  being  tabulated,  form  the  table  of  epochs, 
from  which  the  mean  longitude  of  the  mean  place  of  the  sun  can 
be  found  by  inspection  for  any  day,  hour,  minute  or  second. 

2.  The  Table  of  Longitudes  of  Perigee. — The  longitude  of  peri- 
helion increased  by  180°  is  the  corresponding  longitude  of  perigee. 
Hence  the  former  being  found,  and  its  rate  of  change  determined, 
the  addition  of  180°  to  each  longitude  of  perihelion  will  give  the 
longitude  of  perigee,  and  these  values  being  tabulated  form  the 
table  of  longitudes  of  perigee. 

3.  The  Table  of  Equations  of  the  Center.— The  difference  be- 
tween the  true  and  mean  anomalies  at  any  instant,  given  by  the 
first  of  Eqs.  (650),  Mechanics, 

8  —  n  t  —  2  e  sin  n  t  -f  f  e*  sin  2  n  t  -j-  etc.,  (13) 


6  PRACTICAL  ASTRONOMY. 

is  called  the  Equation  of  the  Center,  and  is  known  when  n  and  0 
are  known ;  t  being  the  time  since  perihelion  passage. 

Assuming  e  to  be  constant  and  causing  n  t  to  vary  from  0°  to 
360°,  the  resulting  values  of  the  second  member  of  the  equation 
will  form  a  table  of  the  equations  of  the  center.  The  errors  in  these 
values  arise  from  the  small  variations  in  the  values  of  e  ;  these 
errors  can  be  found  by  substituting  in  the  second  member  of  the 
above  equation  the  actual  values  of  e  at  the  time,  and  the  differences 
being  talulated  will  give  a  table  by  which  the  equations  of  the 
center  may  be  corrected  from  time  to  time. 

4.  Equation  of  the  Equinoxes  in  Longitude.— Due  to  physical 
causes,  the  pole  of  the  equator  completes  a  revolution  about  the 
pole  of  the  ecliptic  in  about  26,000  years.  The  plane  of  the  equator 
conforming  to  this  motion  of  the  pole,  its  intersection  with  the 
plane  of  the  ecliptic,  called  the  line  of  the  equinoxes,  turns  with  a 
retrograde  motion  of  about  50".2  per  annum  about  the  sun  as  a 
fixed  point. 

This  motion  is  not  however,  perfectly  uniform.  The  true  pole 
describes  once  in- 19  years  around  the  moving  mean  place  above  re- 
ferred to,  a  small  ellipse,  whose  transverse  axis  directed  toward  the 
pole  of  the  ecliptic  is  18".5  in  angular  measure,  and  whose  conju- 
gate axis  is  13".74.  The 'corresponding  irregularity  in  the  motion 
of  the  line  of  the  equinoxes  causes  a  slight  oscillation  of  the  true 
on  either  side  of  the  moving  mean  equinox.  Both  are  on  the  eclip- 
tic; and  their  distance  apart  at  any  time  is  called  the  Equation  of 
the  Equinoxes  in  Longitude,  its  projection  on  the  equator  the 
Equation  of  the  Equinoxes  in  Right  Ascension,  and  the  intersection 
of  the  declination  circle  which  projects  the  mean  equinox  with  the 
equator,  the  Reduced  Place  of  the  Mean  Equinox.  The  maximum 
value  of  the  Equation  of  the  Equinoxes  in  Longitude  is 

i  off  74. 

^4^-  +  sin  23°  28'  =  17".25. 


To  illustrate,  P,  in  Fig.  2,  is  the  pole  of  the  equator,  VE  the 
ecliptic,  VM  the  equator,  F  the  true,  V  the  mean,  and  V"  the  re- 
duced place  of  the  mean  vernal  equinox.  VV  is  the  equation  of 
the  equinoxes  in  longitude,  and  VV"  in  Right  Ascension. 

The  equation  of  the  equinoxes  in  longitude  is  a  function  of  the 


BPHEMERIS.  7 

" 

longitude  of  the  moon's  node,  the  longitude  of  the  sun,  and  the 
obliquity  of  the  ecliptic.     Separate  tables  are  constructed  for  this 


FIG.  2. 

correction,  in  which  the  arguments  for  entering  them  are  the 
obliquity  and  longitude  of  the  moon's  node,  and  the  obliquity  and 
the  longitude  of  the  sun;  the  sum  of  the  two  corrections  is  the  value 
of  the  equation  of  the  equinoxes  in  longitude  at  the  corresponding 
times. 

The  Perturbations  in  Longitude  of  the  earth  arising  from  the 
attractions  of  the  planets  (especially  Venus  and  Jupiter),  are  the 
same  for  the  sun;  these  are  computed  by  the  methods  indicated  in 
Physical  Astronomy,  (see  Art.  174,  Mechanics,)  and  then  tabulated. 

The  Sun's  Aberration  is  taken  to  be  constant,  amounting  to 
—  20".25  and  is  included  in  the  table  of  epochs. 

Ephemeris  of  the  Sun. — The  above  tables  having  been  computed, 
we  proceed  as  follows : 

1.  From  the  table  of  epochs  take  out  the  mean  longitude  of  the 
sun's  mean  place  corresponding  to  the  exact  instant  considered. 

2.  From  the  table  of  longitudes  of  perigee  take  the  mean  longi- 
tude of  perigee;  the  difference  between  this  and  the  mean  longi- 
tude of  the  sun's  mean  place  is  the  mean  anomaly. 

3.  With  the  mean  anomaly  as  an  argument  find  the  correspond- 
ing value  of  the  equation  of  the  center  from  its  table,  and  add  it 


8  PRACTICAL  ASTRONOMY. 


with  its  proper  sign  to  the  mean  longitude  of  the  sun's  mean  place; 
the  result  will  be  the  mean  longitude  of  the  sun's  true  place;  hence 
the 

Sun's  true  longitude  =  Mean  longitude  of  sun's  mean  place 
±  Equation  of  center  ±  Perturbations  in  longitude  ±  Corrections 
to  pass  from  the  mean  equinox  o-f  date  to  true  equinox  of  date. 
These  latter  corrections  are  due  to  Nutation  and  constitute  the 
Equation  of  the  Equinoxes  in  Longitude. 

4.  Having  the  true  longitude  of  the  sun  and  the  obliquity  of  the 
ecliptic,  the  corresponding  Right  Ascension  and  Declination  of  the 
sun  can  be  computed  for  the  same  instant  by  the  method  explained 
in  Art.  180,  Astronomy. 

5.  Earth's  Radius  Vector.  —  Substituting  the  values  of  e  and  n  t, 
in  the  second  of  Eqs.  (650),  Mechanics,  will  give  the  values  of  the 
distance  of  the  sun  from  the  earth  in  terms  of  the  mean  distance 
a:  thus 

(g2 
1  —  e  cos  n  t  +  -  (1  —  cos  2  n  t) 

303  V 

--  —  (cos  3  nt  —  cos  nt)-\-  etc.]  .  (14) 

-  ^ 

6.  The  Sun's  Horizontal  Parallax.  —  From  astronomical  observa- 
tions the  value  of  a  (and  hence  of  r)  is  found  in  terms  of  the  earth's 
equatorial  radius,  pe  .     (Young,  Chapters  XIII  and  XVI.) 

The  sun's  equatorial  horizontal  parallax,  P,  at  any  time  is  then 
given  by 


GO  being  the  number  of  seconds  in  a  radian  =  206264".8,  and  r  being 
expressed  as  just  stated. 

At  any  place  where  the  earth's  radius  in  terms  of  the  equatorial 

radius  is  p,  we  shall  have  for  the  horizontal  parallax  —  =  p  P. 

7.  The  Sun's  Apparent  Semi-Diameter.  —  Knowing  P,  measure- 
ments of  the  sun's  angular  semi-diameter  will  give  its  linear  semi- 
diameter  s'  in  terms  of  pe.  Its  angular  semi-diameter  s  for  any 
day  is  then  given  by 

s  =  Ps'«  (16) 


EPUEME111S.  9 

9.  Equation  of  Time. — If,  at  the  instant  when  the  true  sun's 
mean  place  coincides  with  the  mean  equinox,  an  imaginary  point 
should  leave  the  reduced  place  of  the  mean  equinox  and  travel  with 
uniform  motion  on  the  celestial  equator,  returning  to  its  starting- 
point  at  the  instant  the  true  sun's  mean  place  next  again  coincides 
with  the  mean  equinox,  such  a  point  is  called  a  Mean  Sun.  Time 
measured  by  the  hour  angles  of  this  point  is  called  Mean  Solar 
Time.  The  angle  included  between  the  declination  circles  passing 
through  the  centre  of  the  true  sun  and  this  point  at  any  instant  is 
called  the  Equation  of  Time  for  that  instant;  its  value,  at  any  in- 
stant, added  algebraically  to  mean  or  apparent  solar  time  will  give 
the  other.  As  the  apparent  time  can  be  found  by  direct  observa- 
tion the  equation  of  time  is  usually  employed  as  a  correction  to  pass 
from  apparent  to  mean  solar  time.  Thus  in  Fig.  2,  PM  is  the  me- 
ridian, 8  the  true  sun,  S'  its  mean  place,  S"  the  mean  sun,  VS'" 
the  true  K.  A.  of  the  true  sun,  V"S"  the  mean  R.  A.  of  the  mean 
sun  —  VS'  =  sun's  mean  longitude,  angle  NFS'"  or  arc  MS9" 
apparent  solar  time,  MS"  mean  solar  time,  and  S"Sr"  the  Equation 
of  Time  =  VS'"-(VS"  +  VV"). 

Hence  we  have  for  the  Equation  of  Time, 

e  =  True  sun's  true  Right  Ascension 
—  (sun's  mean  longitude-)-  equation  of  equinoxes  in  R.  A.).  (17) 

The  mean  sun  (S")  moving  in  the  equator  and  used  in  connec- 
tion with  time,  must  not  be  confused  with  the  mean  sun  (S')  before 
referred  to,  moving  in  the  ecliptic. 

10.  Referring  to  the  American  Ephemeris,  we  see  that  Page  I 
of  each  month  contains  the  Sun's  Apparent  R.  A.,  Declination, 
Semi-diameter,  Sidereal  time  of  semi-diameter  passing  the  me- 
ridian, at  Greenwich  apparent  noon,  together  with  the  values  for 
their  respective  hourly  changes;  the  latter  being  computed  from 
the  values  of  their  differential  co-efficients.  From  these  we  can 
find  the  corresponding  data  for  any  other  meridian.  Page  II  con- 
tains similar  data  for  the  epoch  of  Greenwich  mean  moon,  and  in 
addition  the  sidereal  time  or  R.  A.  of  the  mean  sun.  Page  III  con- 
tains the  sun's  true  longitude  and  latitude,  the  logarithm  of  the 
earth's  radius  vector  and  the  mean  time  of  sidereal  noon.  The 


10 


PRACTICAL  ASTRONOMY. 


obliquity,  precession,  and  sun's  mean  horizontal  parallax  for  the 
year,  are  found  on  page  278  of  the  Ephemeris.  All  these  consti- 
tute an  Ephemeris  of  the  Sun. 

From  the  hourly  changes  the  elements  for  any  meridian  can  be 
readily  computed. 


THE   EPHEMERIS   OP   THE    MOON. 

The  Ephemeris  of  the  Moon  consists  of  tables  giving  the  Moon's 
Right  Ascension  and  Declination  for  every  hour  of  Greenwich 
mean  time,  witi  the  changes  for  each  minute;  the  Apparent 
Semi-diameter,  Horizontal  Parallax,  Time  of  upper  transit  on  the 
Greenwich  Meridian,  and  Moon's  Age.  In  order  to  compute  these, 
it  is  first  necessary  to  find  the  True  Longitude  of  the  Moon,  its 
True  Latitude,  the  Longitude  of  the  Moon's  Node,  the  Inclination 
of  the  Moon's  Orbit  to  the  Ecliptic,  and  the  Longitude  of  Perigee. 

1.  The  Elements  of  the  Lunar  Orbit. — Let  DC  be  the  intersection 
of  the  celestial  sphere  by  the  plane  of  the  lunar  orbit;  VB  the 


FIG.  3. 


ecliptic,  and  VA  the .  equinoctial ;  V  the  mean  vernal  equinox,  N 
the  ascending  node,  P  the  Perigee,  all  relating  to  some  assumed 
epoch.  Also  let  Ml ,  M 2 ,  M3 ,  Mt ,  be  the  geocentric  places  of  the 
moon's  center  at  the  four  times,  tl ,  1 3 , 1 3 ,  tt .  These  places  are 


EPHEMERIS.  11 

obtained  as  in  case  of  the  sun  by  observed  Right  Ascensions  and 
Declinations,  corrected  for  refraction,  semi-diameter,  parallax,  and 
perturbations,  then  converted  into  the  corresponding  latitudes  and 
longitudes,  and  finally  referred  to  the  mean  equinox  of  the  epoch, 
by  correcting  for  aberration,  nutation,  and  precession. 
Referring  to  the  figure,  assume  the  following  notation: 

v  =  V  N,  the  longitude  of  the  node; 
t  =  CN  By  the  inclination  of  the  orbit; 
li==  V0iy  the  longitude  of  J/,; 
Z2  =  F02,  the  longitude  of  M^ 
Aj=  Ml  01  ,  the  latitude  of  M^\ 
A2=  M^  0,  ,  the  latitude  of  Jfa; 
v,  =  VEN+  NE  Mlf  the  orbit  longitude  of  J/",; 
p  =  V  EN  -\-  NE  P,  the  orbit  longitude  of  perigee; 
0  =  PE  M^  v1  —  p,  the  true  anomaly  of  Jf,; 
e  —  eccentricity  of  orbit; 
m  =  mean  motion  of  moon  in  its  orbit; 
tl  =  time  since  epoch  for  Ml  ; 
L  =  mean  orbit  longitude  at  epoch. 

To  find  v  and  i,  we  have  from  the  right-angled  spherical  tri- 
angles Ml  N  0,  and  Jf,  N  02 


2  , 


sin  (£j  —  v)  —  cot  i  tan  Aa    / 

sin  (£2  —  v)  =  cot  i  tan  A2    f  '     ' 


and  by  division, 

sin  (?,  -  v)      tanl, 


Adding  unity  to  both  members,  reducing,  then  subtracting  each 
member  from  unity,  again  reducing,  and  finally  dividing  one  result 
by  the  other,  we  obtain 

sin  (Z,  -  r)  +  sin  (7,  -  v)  =  tan  A,  +  tan  A, 

sin  (Z2  —  r)  —  sin  (/,  —  v)      tan  Aa  —  tan  A/  '     ' 

01  by  reduction  formulas,  page  4  (Book  of  Formulas), 


12  PRACTICAL  ASTRONOMY. 

from  which  v  can  be  found;  i  is  found  from  either  of  equations 
(18),  when  v  is  known. 

To  find  L,  m,  e,  and  p,  we  proceed  as  in  the  determination  of 
the  table  of  epochs  in  the  case  of  the  sun,  using  a  similar  equation, 
thus : 

L  -\-  m  T{==  vl  —  2e  sin  (vl  —  p),   } 

£  + *?;  =  *; I  ^££1$;  (32) 

m  Tt  =  v^  —  2  e  sin  (#4  —  p), 


in  which 


..  tan  (L  —  v) 
v  =  v  +  tan  '  -     ±i—.  —  t;  (23) 

cos  *  v    f 


and  similar  values  for  vz  ,  v3  ,  and  v4  . 

To  find  the  ecliptic  longitude  of  perigee  V  0,  represented  by  pl  , 
we  have  from  the  right-angled  triangle  N  P  0, 

tan  N  0  =  tan  (  p  —  v)  .  cos  t,  (24) 

from  which 

pl  =  v  -j-  tan'1  (tan  (p  —  v)  .  cos  i).  (25) 

Similarly  the  mean  ecliptic  longitude  of  the  moon,  Ll  ,  at  the  epoch  is 
L^  =  v+  tan"1  (tan  (L  —  v)  .  cos  t).  (26) 

To  find  the  sidereal  period,  s,  we  have 


-  (27) 

v     7 

in  which  s  is  the  length  of  the  sidereal  period  in  mean  solar  days. 

2.  The  Ephemeris  of  the  Moon.  —  The  motion  of  the  moon  is  much 
more  irregular  and  complicated  than  the  apparent  motion  of  the 
sun,  owing  mainly  to  the  disturbing  action  of  this  latter  body.  But 
this  and  other  perturbations  have  been  computed  and  tabulated, 
and  from  these  tables,  including  those  of  the  node  and  inclination, 
the  places  of  the  moon  in  her  orbit  are  found  in  the  same  way  as 
those  of  the  sun  in  the  ecliptic.  The  mean  orbit  longitude  of  the 
moon  and  of  her  perigee  are  first  found  and  corrected  ;  their  dift'er 
ence  gives  her  mean  anomaly,  opposite  to  which  in  the  appropriate 
table  is  found  the  equation  of  the  center,  and  this  being  applied 


EPHEMERIS.  13 

with  its  proper  sign  to  the  mean  orbit  longitude  gives  the  true  orbit 
longitude,  after  reduction  to  true  equinox  of  date. 

The  Right  Ascension  and  Declination  of  the  Moon  can  now  be 
computed  for  any  instant  of  time,  thus :  subtract  the  longitude  of 
the  node  from  the  orbit  longitude  of  the  moon,  and  we  have  the 
moon's  angular  distance  from  her  node,  represented  in  the  figure  by 
N  Ml .  This,  with  the  inclination  i,  will  give  us  the  moon's  latitude 
and  the  angular  distance  N  0X;  the  latter  added  to  the  longitude 
of  the  node  will  give  the  moon's  longitude  FO,.  The  latitude, 
longitude,  and  obliquity  of  the  ecliptic  suffice  to  compute  the  right 
ascension  and  declination.  The  radius  vector,  equatorial  horizontal 
parallax,  apparent  diameter,  etc.,  are  computed  as  in  the  case  of 
the  sun. 

THE  EPHEMEBIS   OF   A   PLACET. 

From  the  tables  of  a  planet  its  true  orbit  longitude  as  seen  from 
the  sun  is  found,  as  in  the  case  of  the  moon  as  seen  from  the  earth. 
The  heliocentric  longitude  and  latitude,  and  the  radius  vector  are 
found  from  the  heliocentric  orbit  longitude,  heliocentric  longitude 
of  the  node,  and  inclination,  in  the  same  way  as  the  geocentric 
elements  of  the  moon  are  found  from  similar  data  in  the  lunar  orbit. 

To  pass  from  heliocentric  to  geocentric  coordinates,  let  P,  Fig.  4, 
be  the  planet's  center,  E  that  of  the  earth,  S  that  of  the  sun,  and 


FIG.  4. 


0  the  projection  of  P  on  the  plane  of  the  ecliptic.     S  V  and  E  V 
are  drawn  to  the  vernal  equinox;  then  let 


14  PRACTICAL  ASTRONOMY. 

r   =  E  S,  be  the  earth's  radius  vector; 

/•'  =  S  P,  be  the  planet's  radius  vector; 

X   =  V  S  0,  be  the  heliocentric  longitude  of  planet; 

A'  =  VE  0,  be  the  geocentric  longitude  of  planet; 

Q    —  P  S  0,  be  the  heliocentric  latitude  of  planet; 

8'  =  P  E  0,  be  the  geocentric  latitude  of  planet; 

S  =  0  S  E,  be  the  commutation  ; 

0  =  S  0  E,  be  the  heliocentric  parallax; 

E  =  8  E  0,  be  the  elongation; 

L  =  V  E  S,  be  the  longitude  of  the  sun; 

r"=  E  P,  be  the  distance  of  planet  from  the  earth. 

To  find  the  geocentric  longitude, 

S0  =  r'  cos  0,  (28) 

VST=  FJ0tf  =  360°  -  L,  (29) 

S  -  i8o0  _  (360°  -  L)  -  A  =  L  -  180°  -  A,  (30) 

from  which  S  is  known. 

In  the  plane  triangle  0  E  S,  we  have 


0).  (31) 

^=180°,  (32) 

)  =  90°-f,  (33) 

hence 

tani(^-0)^cot^rr;Cc°oSJ-;,  (34) 

and  placing 


we  have 

tan  }  (E  -  0)  =  cot  \  8  ^  =  cot  i  -S  tan  (^  -  45°)        (36) 


INTERPOLATION:  15 

therefore  E  and  0  are  known :  and  we  have 

A'  =  E  -  (360°  -L)=E+L-  360°.  (37) 

To  find  the  geocentric  latitude,  we  have 

P  0  =  ^  0  tail  0'  =  S  0  tan  0  (38) 

tanfl'  _     ff  0  _    jdnj? 
tan  0   '~'  E0~~~   sin~#; 
whence 

at  n  sin  ^ 

tatr-itatff-jg-g.  (40) 

To  find  r",  we  have 

E0  =  r"  cos  0', 
S0  =  r'  cos  0. 

In  the  triangle  E  S  0,  we  have 

r"  cos  0' :  r'  cos  6  ::  sin  $  :  sin  E, 


whence 


,  cosfl  sin/Sf 

~r  ~ 


With  these  data  we  can  readily  find  the  right  ascension,  decli- 
nation, horizontal  parallax,  and  apparent  diameter  as  in  the  case  of 
the  sun  and  moon. 


INTERPOLATION. 

Interpolation. — Whenever  the  differences  of  the  quantities  re- 
corded in  the  Ephemeris  tables  are  directly  proportional  to  the  dif- 
ferences of  the  corresponding  times,  simple  interpolation  will  enable 
us  to  find  the  numerical  value  of  the  quantity  in  question.  When 
this  is  not  the  case,  the  value  is  determined  by  the  "  method  of  in- 
terpolation by  differences."  Bessel's  form  of  this  formula,  usually 
employed,  is 


16 


PRACTICAL  ASTRONOMY. 


n   n  ~ 


n   n  ~ 


(n-\ 

T  ' 

-i): 

n  (n 

i 

i) 

(»- 

-2)n 

^4  + 

etc. 

2.3, 

,4 

(42) 


In  this  formula,  Fn  is  the  value^pf  the  function  to  be  deter- 
mined; F,  the  ephemeris  value  from  which  we  set  out;  dl9d^}  d3, 
etc.,  are  the  terms  of  the  successive  orders  of  differences,  deter- 
mined as  explained  below;  n  is  the  fractional  value  of  the  time 
interval,  in  terms  of  the  constant  interval  taken  as  unity  corre- 
sponding to  which  the  values  of  the  function  F  are  computed  and 
recorded  in  the  tables.  To  use  this  formula,  draw  a  horizontal 
line  below  the  value  of  F  from  which  we  set  out,  and  one  above 
the  next  consecutive  value  taken  from  the  ephemeris.  These  lines 
are  to  enclose  the  values  of  the  odd  differences  diyd3,d6,  etc.  The 
values  of  the  even  differences  d^ ,  dt ,  d6 ,  etc.,  being  each  the  mean 
of  two  numbers,  one  above  and  one  below  in  their  respective  col- 
umns, are  then  inserted  in  their  proper  places.  The  following  ex- 
ample is  given  to  illustrate  the  application  of  Bessel's  formula. 

Find  the  distance  of  the  moon's  center  from  Regulus  at  9  P.M. 
West  Point  mean  time  March  24th,  1891. 

The  longitude  of  West  Point  is  4.93  hrs.  west  of  Greenwich; 
hence  the  Greenwich  time  corresponding  to  9  P.M.  West  Point 
mean  time  is  13.93  hrs.  Referring  to  pages  54  and  55  American 
Ephemeris  we  take  out  the  following  data,  namely : 


A 
March  24. 

F 

*, 

d> 

^3 

*, 

6h 

27°  01'  24" 

1°  28'     9" 

9h 

28°  29'  33" 

+  11" 

1°  28'  20" 

+  1" 

12h 

29°  57'  53" 

+  12" 

9" 

—  •    £j 

13'  .93 

30°  54'  47".31 

1°  28'  32" 

(+  11".5) 

-1" 

-1" 

15h 

31°  26'  25" 

+  11" 

0 

1°  28'  43" 

-1" 

18" 

32°  55'     8" 

+  10" 

1°  28'  53" 

21h 

34°  24'     1" 

THE  TRANSIT.  17 

Whence,  substituting  in  the  formula,  we  have 

^=29°  57'  53"  +  0.643  (1°  28'  32") -}- 0.643    (— ^i1)    (11".5) 

+  (0.643)  §  (  -  0.357)  £  (0.143)  (-  1"). 
=  29°  57'  53"  +  56'  55".616  -  1".32  +  0".01, 
=  30°  54'  47".31  the  required  distance. 

Instruments. — The  principal  instruments  used  in  field  astronom- 
ical work  are  the  Transit,  Sextant,  Zenith  Telescope,  and  Altazi- 
muth or  Astronomical  Theodolite.  A  short  description  of  each 
instrument  will  be  given  in  connection  with  the  first  problem  in- 
volving its  use.  But  since  much  relating  to  the  transit  is  appli- 
cable also  to  the  zenith  telescope  and  altazimuth,  that  instrument 
will  be  explained  first. 

THE  TKANSIT. 

The  Transit  is  an  instrument  usually  mounted  in  the  meridian, 
and  employed  in  connection  with  a  chronometer  for  observing  the 
meridian  passage  of  a  celestial  body.  Since  the  E.  A.  of  a  body  is 
equal  to  the  sidereal  time  at  the  instant  of  its  meridian  passage,  or 
is  equal  to  the  chronometer  time  plus  its  error  (a  =  T  -{-  E),  it  is 
seen  that  by  noting  T,  E  will  be  given  when  a  is  known,  and  con- 
versely a  will  be  given  when  E  is  known.  The  very  accurate 
determination  of  E  is  the  chief  use  of  the  transit  in  field  work. 

The  instrument  consists  essentially  of  a  telescope  mounted  upon 
and  at  right  angles  to  an  axis  of  such  shape  as  to  prevent  easy 
flexure.  The  ends  of  this  axis  called  the  pivots,  areviisually  of 
hard  bell  metal  or  polished  steel,  and  should  be  portions  of  the 
same  right  cylinder  with  a  circular  base.  They  rest  upon  Y's, 
which  in  turn  are  supported  by  the  metal  frame  or  stand.  At  one 
end  of  the  axis  there  is  a  screw  by  which  its  Y  may  be  slightly 
raised  or  lowered  in  order  that  the  axis  may  be  made  horizontal. 
At  the  other  end  of  the  axis  is  another  screw  by  which  its  Y  may 
be  moved  backward  or  forward,  in  order  that  the  telescope  may  be 
placed  in  the  meridian.  The  telescope  is  provided  with  an  achro- 
matic object  glass,  at  the  principal  focus  of  which  is  a  wire  frame 
carrying  an  odd  number  of  parallel  vertical  wires  as  symmetrically 
disposed  as  possible  with  reference  to  the  middle;  also  two  horizon- 
tal wires  near  to  each  other,  between  which  the  image  of  the  point 


18 


PRACTICAL  ASTRONOMY. 


FIG.  5.— THE  TRANSIT. 


THE  TRANSIT.  19 

observed  should  always  be  placed.  This  system  of  wires  is  viewed 
by  a  positive  or  Ramsden's  eye-piece,  which  can  be  moved  bodily 
in  a  horizontal  direction  to  a  position  directly  opposite  any  wire, 
thus  practically  enlarging  the  field  of  direct  view.  The  wires  are 
rendered  visible  in  the  daytime  by  the  diffuse  light  of  day,  but  at 
night  artificial  illumination  is  required.  This  is  effected  by  passing 
light  from  a  small  lamp  along  the  length  of  the  perforated  axis, 


FIG.  6. 

whence  it  is  thrown  toward  the  eye  by  a  small  reflector  placed  at 
the  junction  of  the  axis  and  the  telescope  tube,  thus  producing  the 
effect  of  "  a  bright  field  and  dark  wires." 

The  right  line  passing  through  the  optical  center  of  the  object 
glass  intersecting  and  at  right  angles  to  the  axis  of  rotation  of  the 
instrument,  is  called  the  "line  of  collimation." 

The  wire  frame  should  be  so  placed  that  this  line  will  pass  mid- 
way between  the  two  horizontal  wires,  and  intersect  the  middle 
vertical  wire;  which  latter  should  also  be  at  right  angles  to  the  axis 
of  rotation  of  the  instrument. 

These  conditions  being  fulfilled,  it  is  manifest  that  if  the  axis 
be  placed  in  a  true  east  and  west  line,  and  be  made  exactly  level, 
the  line  joining  any  point  of  the  middle  wire  and  the  optical  center 
of  the  objective  will,  as  the  instrument  is  turned  on  its  pivots, 
trace  on  the  celestial  sphere  the  true  meridian;  and  the  sidereal 
time  when  any  body  appears  on  the  middle  wire,  will,  if  correctly 
estimated,  be  the  value  of  T  required  in  the  equation, 

<x=  T  +  E. 


20  PRACTICAL  ASTRONOMY. 

The  improbability  of  estimating  T  with  precision  leads  to  the 
use  of  more  than  one  wire,  although  the  advantage  of  increasing  the 
number  beyond  five  is,  according  to  Bessel,  very  slight.  If  th$ 
wires  are  grouped  in  perfect  symmetry  with  reference  to  the  mid- 
dle, evidently  the  mean  of  the  times  when  a  star,  as  it  passes  across 
the  field  of  view,  is  bisected  by  each  wire  will  give  a  more  trust- 
worthy time  of  meridian  passage  Mian  if  a  single  wire  be  used. 
Even  if  they  are  not  grouped  in  perfect  symmetry,  the  same  will  be 
true,  after  applying  a  correction  deduced  from  the  "  Equatorial 
Intervals  "  to  be  explained  hereafter.  Every  transit  instrument  is 
provided  with  a  level,  a  diagonal  eye-piece,  one  or  more  setting 
circles,  and  usually  with  a  R.  A.  micrometer.  In  the  case  of  field 
transits  a  striding  level  is  generally  used.  Its  feet  are  provided 
with  Y's  which  are  placed  on  the  pivots  of  the  instrument.  Before 
using,  it  should  be  put  in  adjustment  according  to  the  principles 
explained  in  connection  with  surveying  instruments. 

The  diagonal  eye-piece  facilitates  the  observation  of  stars  near 
the  zenith  by  reflecting  the  rays  at  right  angles  after  they  pass  the 
wires. 

The  setting  circles  are  firmly  attached  to  the  telescope  tube  and 
are  read  by  an  index  arm  carrying  a  vernier,  to  which  is  also  attached 
a  small  level.  They  may  be  arranged  to  point  out  the  position  of 
a  star  either  by  its  declination  or  its  meridian  altitude.  In  the 
latter  case,  the  altitude  is  computed  by  the  formula 

Mer.  Alt.  =  Dec.  +  Co-Latitude, 
for  stars  south  of  the  zenith,  and  by 

Mer.  Alt.  =  Latitude  ±  Polar  Distance, 

for  stars  north  of  the  zenith,  the  upper  sign  being  used  for  stars 
above  the  pole.  In  any  case  having  determined  the  "  setting,"  place 
the  index  arm  to  mark  it,  and  turn  the  instrument  on  its  pivots 
until  the  bubble  plays.  The  star  will  appear  to  pass  through  the 
field  from  west 'to  east,  except  in  case  of  sub-polars,  which  move 
from  east  to  west.  An  equatorial  star  passes  through  the  field  with 
considerable  velocity,  only  40  to  60  seconds  being  required  for  its 
passage,  the  apparent  path  being  a  right  line.  For  other  stars  the 


THE  TRANSIT.  21 

time  required  is  greater,  and  the  path  becomes  more  curved,  until 
as  we  approach  the  pole  several  minutes  are  required,  and  the  cur- 
vature becomes  very  apparent. 

These  facts  are  of  importance  in  determining  when  and  where 
tf  look  for  the  star. 

The  curvature  of  path  must  be  considered  in  determining  the 
r  Equatorial  Intervals."  The  eye-piece  should  be  moved  horizon- 
tally,; keeping  pace  with  the  star,  presenting  the  latter  always  in 
the  middle  of  the  field  of  view. 

The  uses  of  the  R.  A.  micrometer  will  be  explained  hereafter. 

ADJUSTMENTS   OF   THE   TEAKSIT. 

From  the  above  it  is  manifest  that,  assuming  the  objective  to 
be  properly  adjusted,  there  are  five  adjustments  to  be  made  before 
the  instrument  is  ready  for  use. 

1.  To  Place  the  Wires  in  the  Principal  Focus  of  the  Objective. — 
Push  in  or  draw  out  the  eye-piece  till  the  wires  are  seen  with  perfect 
distinctness,  using  an  eye-piece  of  high  power.  Direct  the  telescope 
to  a  small  well-defined  terrestrial  object,  not  nearer  than  two  or 
three  miles.     Now  if  the  wires  are  not  in  the  focus  of  the  objective, 
the  object  will  appear  to  move  with  reference  to  the  wire  as  the  eye 
is  moved  from  side  to  side. 

The  wire  frame  must  then  be  carried  slightly  toward  or  from 
tne  objective  until  this  parallax  is  corrected. 

After  the  instrument  has  been  placed  in  the  meridian,  and  the 
horizontal  wire  made  truly  horizontal,  as  explained  in  the  following 
adjustments,  let  an  equatorial  star  run  along  the  wire,  and  if  it  does 
not  remain  accurately  bisected  while  the  eye  is  moved  up  and  down, 
the  wires  are  not  exactly  in  the  principal  focus.  Other  stars  must 
then  be  used  until  the  parallax  is  removed.  The  wires  are  then  at 
the  common  focus  of  the  objective  and  eye-piece. 

2.  To  Level  the  Axis. — The  striding  level  is  usually  graduated 
from  the  center  toward  each  end. 

The  pivots  are  assumed  to  be  equal. 

If  when  its  Y's  are  applied  to  the  transit  pivots,  the  axis  of  the 
tube  is  parallel  to  the  axis  of  the  pivots  *  (i.e.,  if  the  level  be  in 

*  The  axis  of  the  tube  is  of  course  a  circular  arc  of  long  radius.  Strictly 
speaking,  it  is  the  chord  ot  this  arc  which,  when  the  level  is  perfectly  adjusted, 
will  be  parallel  to  the  axis  of  the  pivots. 


22  PRACTICAL  ASTRONOMY. 

perfect  adjustment),  and  if  w  and  e  denote  the  readings  of  the  west 
and  east  ends  of  the  bubble  respectively,  then 


will  denote  the  reading  of  the  micelle  of  the  bubble,  and  will  there- 
fore measure  the  inclination  of  the  axis  of  the  pivots  in  level  divi- 
sions. But  the  accurate  adjustment  of  the  level  is  never  to  be  as- 
sumed. If  the  axis  of  the  level  be  inclined  to  the  axis  of  the  pivots 
by  such  an  amount  as  to  increase  the  west  reading  and  therefore 
diminish  the  east  reading  by  x  divisions,  then  w  and  e  still  denoting 
the  actual  readings,  we  shall  have  for  the  true  inclination  of  the 
axis  of  the  pivots, 

w  —  e      2x 
2         'IT 

Upon  reversing  the  level,  the  west  and  east  readings  will  be  as  much 
too  small  and  too  large  respectively  as  they  were  too  large  and  too 
small  before  reversal;  therefore  w'  and  e'  denoting  the  actual  read- 
ings, we  shall  have  for  the  true  inclination  this  second  value, 

w'  —  e'      2x 

~2~      '    2' 


The  mean  of  these  two  values, 

(w-e)  +  (w9  -  e') 

2 


is  expressed  only  in  actual  level  readings  and  is  free  from  x,  the  un- 
known effect  of  maladjustment  of  level. 

Hence  to  level  the  axis  —  Take  direct  and  reverse  readings  with 
the  level,  altering  the  inclination  of  the  axis  till  the  sum  of  the 
west  equals  the  sum  of  the  east  readings. 

If  the  level  be  graduated  from  end  to  end,  a  similar  discussion 
will  show  the  level  error  to  be 

(W  +  e)  -  (w'  +  e') 


THE  TRANSIT.  23 

3.  To  Place  the  Wires  at  Right  Angles  to  the  Rotation  Axis. 

— Bisect  a  very  distant  small  terrestrial  object  by  the  middle  wire, 
and  the  axis  being  level,  note  whether  the  bisection  remains  perfect 
from  end  to  end  of  the  wire  as  the  telescope  is  alternately  elevated 
and  depressed.  If  not,  rotate  the  box  carrying  the  wire  frame, 
until  the  above  condition  is  fulfilled. 

The  side  wires  are  parallel  and  the  horizontal  wires  perpendicu- 
lar, to  the  middle  wire. 

After  the  instrument  has  been  finally  placed  in  the  meridian, 
this  adjustment  must  be  verified  by  noting  whether  an  equatorial 
star  will  remain  accurately  bisected  by  the  horizontal  wire  during 
its  passage  through  the  field. 

4.  To  Place  the  Middle  Wire  in  the  Line  of  Collimation.— Bisect 
the  same  distant  object  as  before.   Lift  the  telescope  carefully  from 
the  Y's  and  replace  it  with  the  axis  reversed.      If  the  object  is  still 
perfectly  bisected  the  collimation  adjustment  is  complete.     If  not, 
move  the  wire  frame  laterally  by  the  proper  screws  over  an  estimated 
half  of  the  distance  required  to  reproduce  bisection.     If  the  half 
distance  has  been  correctly  estimated,  the  middle  wire  is  now  in  the 
line  of  collimation.    Repeat  the  operation  from  the  beginning  until 
the  condition  is  fulfilled. 

If  a  proper  terrestrial  point  can  not  be  obtained,  the  cross- wires 
in  an  ordinary  surveyor's  transit  or  theodolite  adjusted  to  stellar 
focus,  will  answer  quite  as  well.  If  two  theodolites  are  placed,  one 
north  and  the  other  south  of  our  transit,  pointing  toward  and 
accurately  adjusted  on  each  other,  the  reversal  of  the  axis  above 
referred  to  may  be  avoided. 

In  all  these  cases,  the  R.  A.  micrometer  is  of  great  convenience 
for  measuring  the  distance  whose  half  is  to  be  taken. 

The  parts  of  the  instrument  are  now  in  adjustment  among  them- 
selves. It  remains  to  adjust  the  instrument  as  a  whole  with  refer- 
ence to  the  celestial  sphere;  i.e.,  to  so  place  the  instrument  that 
when  turned  on  its  pivots,  the  line  of.  collimation  shall  trace  the 
true  meridian. 

5.  To  Place  the  Line  of  Collimation  in  the  Meridian.— This  is 
most  easily  effected  by  the  aid  of  a  sidereal  chronometer  whose  error 
is  known.     The  instrument  is  first  placed  as  nearly  in  the  proper 
position  as  can  be  estimated,  and  its  supporting  frame  turned  in 


24  PRACTICAL  ASTRONOMY. 

azimuth  until  the  telescope  can  be  pointed  at  a  slow  moving  star  at 
about  the  time  of  its  meridian  passage. 

Now  level  the  axis  carefully,  set  the  telescope  to  the  meridian 
altitude  of  a  circum-polar  star  whose  place  is  given  in  the  Ephem- 
eris,  and  bring  the  middle  vertical  wire  upon  this  star  a  short  time 
before  its  meridian  passage, ,  Hold  £he  wire  upon  the  moving  star 
by  turning  the  screw  which  moves  one  of  the  Y's  in  azimuth,  until 
the  chronometer  corrected  for  its  error  indicates  a  time  equal  to  the 
star's  R.  A.  for  the  date.  The  transit  is  now  very  approximately  in 
the  meridian,  although  the  adjustment  should  be  tested  by  other  stars. 

Since  the  observations  to  be  made  with  the  transit  will  be  for 
the  purpose  of  an  accurate  determination  of  the  chronometer  error, 
this  latter  will  usually  be  known  only  approximately.  It  may  how- 
ever be  found  with  sufficient  accuracy  for  making  the  adjustment 
by  noting  that  since  all  vertical  circles  intersect  at  the  zenith,  the 
time  of  a  zenith  star's  passage  over  the  middle  wire  will  be  its  time 
of  passage  over  the  meridian  even  though  the  transit  be  not  in  the 
meridian.  The  difference  between  the  chronometer  time  of  this 
event  and  the  star's  R.  A.  will  therefore  be  the  clock  error. 

In  the  absence  of  a  zenith  star,  two  circum-zenith  stars,  at  op- 
posite and  nearly  equal  zenith  distances,  will  give  values  of  the  clock 
error  differing  about  equally  and  in  opposite  directions  from  its  true 
value. 

Alternating  observations  on  circnm -polar  and  circum-zenith 
stars  will  now  give  the  required  adjustment  with  two  or  three 
trials. 

As  a  final  test,  the  values  of  the  chronometer  error  determined 
from  stars  which  cross  the  meridian  at  widely  separated  points 
should  be  practically  identical. 

INSTRUMENTAL  CONSTANTS. 

These  must  be  determined  before  the  instrument  can  be  used, 
and  are  five  in  number,  the  transit  is  supposed  to  be  in  good 
adjustment. 

1.  The  Value  in  Time  of  One  Division  of  the  R.  A.  Micrometer 
Head. — The  micrometer  head,  which  is  usually  divided  into  100 
equal  parts,  carries  a  movable  wire  which  is  always  parallel  to  the 
fixed  vertical  wires  of  the  transit,  and  as  nearly  as  possible  in  their 


INSTRUMENTAL   CONSTANTS.  25 

plane.  As  it  moves  across  the  field  of  view  it  apparently  coincides 
with  each  of  them  in  succession. 

If  s  denote  the  angular  distance,  measured  from  the  optical 
center  of  the  objective,  between  two  positions  of  the  micrometer 
wire,  one  of  which  coincides  with  the  middle  wire  or  the  meridian 

a 

of  the  instrument,  then  —  —  i  will  be  the  interval  of  time  required 
-Lo 

for  a  star  exactly  on  the  celestial  equator  to  pass  from  one  position 
to  the  other;  since  it  is  only  such  stars  whose  diurnal  path  is  a 
great  circle,  and  since  also  intervals  of  time  are  measured  by  area 
of  a  great  circle — the  equator. 

"With  a  star  exactly  on  the  equator,  the  process  of  finding  the 
value  of  one  division  of  the  R.  A.  micrometer  head  would  therefore 
consist  in  noting  the  time  required  for  the  star  to  pass  from  one 
position  of  the  wire  to  the  other;  the  quotient  of  which  by  the 
number  of  turns  or  divisions  through  which  the  head  has  been 
moved  would  give  the  value  of  one  turn  or  division. 

In  the  absence  of  such  a  star  we  must  select  one  whose  declina- 
tion, 6,  is  not  zero.  The  interval  of  tkne  required  for  such  a  star 
to  pass  from  one  position  to  the  other  will  be  given  by  the  equation 

sin  7  =  sin  i  sec  d.  (43) 

To  prove  this,  let  A  B  in  Figure  7,  which  represents  the  sphere 
projected  on  the  plane  of  the  horizon,  be  the  meridian,  P  the  pole, 

A 


E  Q  R  the  equator,  S  the  place  of  the  star,  MZ  the  first  position 
of  the  wire,  and  P  Ft  coinciding  with  the  meridian,  the  second. 


26  PRACTICAL  ASTRONOMY. 

Through  8  pass  an  arc  of  a  great  circle,  K  S,  perpendicular  to 
A  B.  This  arc  will  be  equal  to  Q  L,  and  will  therefore,  from  what 
precedes,  be  denoted  by  s. 

Hence,  in  the  right-angled  triangle  SP  K,  we  have 


But  P  is  the  hour  angle  of  the  star  at  8,  and  s  is  the  hour  angle  of 
an  equatorial  star  at  an  equal  angular  distance  from  the  meridian, 
i.e.,  at  L. 

Hence  denoting  the  time  equivalent  of  the  former  by  /,  and  of 
the  latter  by  i  as  before,  we  have 

sin  /=  sin  i  sec  d, 
and  therefore 

.sin  i  =  sin  /cos  d.  (#) 

From  this  equation  we  inay  compute  i,  8  being  taken  from  the 
Ephemeris,  and  1;  which  is  directly  observed,  being  the  sidereal 
time  required  for  the  star  to  pass  from  S  to  the  meridian. 

After  which,  if  R  denote  the  value  of  a  revolution  or  division 
of  the  micrometer  head,  and  N  the  number  of  revolutions  or  divi- 
sions corresponding  to  /,  we  have  for  the  value  in  time 


If  the  star  be  not  within  10°  of  the  pole  we  may  write 

i  —  I  cos  #, 


and 


~, 


thus  avoiding  the   "  Correction   for    Curvature "  involved  in  the 
trigonometric  functions. 

By  examining  the  equations 

• 

sin  i  —  sin  7  cos  tf ,  and  i  —  I  cos  3,  (45) 


INSTRUMENTAL   CONSTANTS.  27 

it  is  seen  that  for  the  accurate  determination  of  i,  it  is  better  to  use 
stars  near  the  pole,  since  errors  in  the  observed  values  of  /  will 
then  be  multiplied  by  the  cosine  of  an  angle  near  90°. 

Therefore,  to  determine  this  constant,  proceed  as  follows: 

Shortly  before  the  time  of  culmination  of  some  slow-moving 
(circum-polar)  star  set  the  instrument  so  that  the  star  will  pass 
through  the  field.  Set  the  micrometer  head  at  some  exact  division, 
with  the  wire  on  the  side  of  the  field  where  the  star  is  about  to 
enter.  Note  the  reading  of  the  micrometer  head,  and  record  the 
time  of  passage  of  the  star  over  the  wire,  using  a  sidereal  chronom- 
eter whose  rate  is  well  determined.  Set  the  wire  again  a  short 
distance  ahead  of  the  star,  note  the  reading,  and  record  the  time  of 
passage.  In  this  manner  "step"  the  screw  throughout  its  entire 
length.  Then,  remembering  that  /  is  the  sidereal  interval  (cor- 
rected for  rate  if  appreciable)  between  any  given  passage  and  that 
obtained  when  the  wire  was  nearest  to  the  meridian  or  the  center 
of  the  field  of  view,  apply  to  each  pair  of  observations  equations 
(a)  and  (b),  or  (c)  and  (d),  according  to  the  value  of  o\ 

Where  d  is  considerably  less  than  90°  and  equations  (c)  and  (d) 
are  used,  the  correction  for  curvature  of  path  becomes  very  small, 
and  the  same  necessity  does  not  exist  for  comparing  each  observa- 
tion with  the  one  made  at  the  center  of  the  field. 

No  correction  for  difference  of  refractions  between  any  two 
positions  of  the  star  is  required,  since  at  its  meridian  passage  the 
star  is  moving  almost  wholly  in  azimuth. 

In  any  case  the  adopted  value  of  the  constant  should  rest  on 
many  such  determinations. 

Very  convenient  stars  4o  use  are  a,  d,  ft,  Ursae  Minoris.  Their 
decimations  are  accurately  given  in  the  Ephemeris,  the  first  two 
for  every  day,  and  the  last  one  for  every  ten  days. 

The  first  two  require  equations  (a)  and  (b). 

The  last  one  not  necessarily  so. 

2.  The  Equatorial  Intervals. — By  the  "  Equatorial  Interval "  of 
a  given  wire  is  meant  the  interval  of  sidereal  time  required  for  a 
star  on  the  celestial  equator  to  pass  from  this  wire  to  the  middle 
wire,  or  vict  versa. 

The  method  of  determinating  this  constant  for  each  wire  is 
manifestly  identical  in  principle  with  the  process  just  described, 
omitting  the  application  of  equations  (b)  or  (d),  and  remembering 


28  PRACTICAL  ASTRONOMY. 

that  /  is  the  observed  interval  with  a  star  whose  declination  is  tf, 
and  i  is  the  required  Equatorial  Interval. 

Another  method,  which  may  either  be  used  independently  or  as 
a  verification,  is  to  measure  the  intervals  between  the  wires  (in 
time)  by  the  E.  A.  micrometer.  The  adopted  constants  should  rest 
upon  many  determinations. 

3.  The  Reduction  to  the  Middle  Wire.—  The  mean  of  the  times 
of  transit  of  a  celestial  body  over  the  several  wires  of  a  transit  in- 
strument is  called  the  time  of  transit  over  the  mean  of  the  wires 
or  the  mean  wire.  The  mean  does  not  usually  coincide  with  the 
middle  wire,  due  to  the  improbability  of  grouping  the  wires  in  per- 
fect symmetry  with  reference  to  the  middle. 

Since  it  is  the  middle  wire  which  has  been  placed  in  the  merid- 
ian, it  becomes  necessary  to  determine  the  distance,  in  time,  of  the 
mean  from  the  middle  wire.  Then,  the  mean  of  the  times  of  tran- 
sit being  corrected  by  this  constant,  we  will  have  a  very  accurate 
determination  of  the  time  of  transit  over  the  meridian.  Suppose 
the  instrument  to  have  seven  wires,  and  to  be  in  good  adjustment. 
A  star  at  its  upper  culmination  will  apparently  move  over  these 
wires  from  west  to  east;  therefore  (with  the  instrument  in  a  given 
position,  say  with  "  illumination  east  ")  let  the  wires  be  successive!}7 
numbered  from  the  west  towards  the  east. 

Let  a  star  whose  declination  is  d  pass  through  the  field,  and  let 
*i  jti>t*  >  *4  »  #6  >  ^  .'  t,  >  be  tne  accurate  instants  of  passing  the  cor- 
responding wires;  let  i,  ,  ?\  ,  iz  ,  0,  ?'fi  ,  *6,  i,  ,  be  the  equatorial  inter- 
vals from  the  middle  wire.  Then  the  time  of  passing  the  mean 
wire  is 


(46) 


The  time  of  passing  the  middle  wire  is  either 


tl  +  il  sec  d,  tz  -f  i9  sec  d,  tz  -f  i3  sec  d,  tt,tf-  ^  sec  tf,  t6  —  i6  sec  tf, 

or  t^  —  *7  sec  d 

(note  the  minus  sign  in  the  last  three).      Hence  the*  most  probable 
time  of  passing  the  middle  wire  is 

ft 

2t  t   Si 
=      -+secd.  (47) 


INSTRUMENTAL  CONSTANTS,  29 

The  difference  between  this  and  the  time  of  passing  the  mean 
wire  is  evidently  the  second  term,  or 

^  sec  6  =  (*!  +  *• +  *«)"•(*»+*•  +  *''  sec  tf.  (48) 

i  • 

The  equatorial  value  of  this  reduction  (the  desired  constant) 
will  then  be 


and  for  any  given  star  the  actual  reduction  will  be  this  value  mul- 
tiplied by  sec  d.  The  adopted  value  of  A  i  should  rest  upon  many 
determinations.  Its  sign  is  evidently  changed  by  reversing  the  axis 
of  the  instrument. 

Hence,  to  find  the  time  of  a  star's  passage  over  the  middle  wire, 
we  have  the  rule :  To  the  mean  of  the  times  add  A  i  sec  d,  noting 
the  signs  of  both  factors. 

The  Equatorial  Intervals  are  also  used  for  finding  the  time 
of  passage  over  the  middle  wire  when  actual  observation  on  some  of 
the  wires  has  been  prevented  by  clouds  or  other  cause.  Thus  suppose 
observations  have-only  been  made  on  the  second,  third,  and  seventh 
wires.  The  most  probable  time  of  passing  the  middle  wire  is 

(*.  +  f.  sec  (?)  +  (*.  +  i*  sec  <?)  +  (*T  -  *T  sec  6}  _  2 1       2i 

~3~  T        ~3~  '    '  d> 

t  and  i  referring  only  to  the  wires  used. 

4.  Value  of  One  Division  of  the  Level. — In  practical  astronomy 
the  level  is  used  not  merely  for  testing  and  regulating  the  horizon- 
tality  of  a  given  line,  but  also  for  measuring  either  in  arc  or  time 
those  small  residual  inclinations  to  the  horizontal  which  no  process 
of  mechanical  adjustment  can  either  eliminate  or  maintain  at  a 
constant  value. 

Hence  we  must  determine  the  value  of  one  division  of  the  strid- 
ing level  of  the  transit;  i.e.,  the  increment  or  decrement  of  incli- 
nation which  will  throw  the  bubble  one  division  of  the  gradu  nfcion. 

The  best  method  of  determining  this  quantity  in  case  of  a  de- 
tached level  is  by  use  of  tne  "  Level-trier,"  which  consists  simply 
of  a  metal  bar  resting  at  one  end  on  two  firm  supports,  and  at  the 


30  PRACTICAL  ASTRONOMY. 

i 

other  on  a  vertical  screw.  Then  if  d  be  the  distance  from  the  screw 
to  the  middle  of  the  line  joining  the  two  fixed  supports,  and  h  the 
distance  between  two  threads  of  the  screw  (obtained  by  counting 
the  number  of  threads  to  the  inch),  the  inclination  of  the  bar  to  the 

horizon  would  be  changed  by  -T-  v-^?ar/>  due  to  one  revolution  of  the 

d  sin  J_ 

screw.  The  level  is  then  placed  on  the  bar  and  the  number  n  of 
divisions  passed  over  by  the  bubble  due  to  one  turn  (or  division)  of 
the  screw  is  noted.  The  value  of  one  division  of  the  level  in  angle 

is  then  —  =—  :  —  777  .     The  mean  of  several  observations,  using  both 
ndsm  1" 

ends  of  the  bubble,  should  be   adopted.      The  value  in  time  is 

—  -  j—  -.  —  —r-.  .  If  no  level-trier  is  available,  the  level  should  be 
15  n  d  sin  I" 

placed  on  the  body  of  the  telescope  connected  with  a  vertical  circle 
reading  to  seconds  :  as  for  example  the  meridian  circle  of  a  fixed 
observatory.  Move  the  instrument  slowly  by  the  tangent  screw  and 
note  the  number  of  level  divisions  corresponding  to  a  change  of  1" 
in  the  reading  of  the  circle,  taking  the  means  as  before.  By  either 
method  the  level  may  be  tested  throughout  its  entire  length. 

We  have  seen  that  the  inclination  of  a  line  in  level  divisions^  is 

(w  -f-  w')  —  (e  -f  #')     i  -f   r>  j  L   -     L  f       j 

—  !  -  —  —  —  ;  hence  if  D  denote  the  constant  just  found, 

the  inclination  of  the  line  in  arc  will  be 


p_  -g     e  _  - 

~~  •"-  ~~ 


the  west  end  being  higher  if  (w  -f-  w')  >  (e  -j-  e'),  or  when  this  ex- 
pression is  positive. 

5.  Inequality  of  the  Pivots.  —  The  construction  of  the  pivots 
being  one  of  the  most  delicate  operations  in  the  manufacture  of  the 
whole  instrument,  their  equality  must  never  be  assumed. 

In  transit  observations  it  is  manifestly  the  axis  of  rotation  (the 
Axis  of  the  pivots)  which  should  be  made  horizontal,  or  whose  in- 
clination should  be  measured.  If  the-  pivots  are  unequal  they  may 
be  regarded  as  portions  of  the  same  right  cone;  in  which  case  it  is 
evident  that  the  striding  level  applied  to  the  upper  element  might 
indicate  horizontality  when  the  axis  was  really  inclined,  and  vice 


INSTRUMENTAL  CONSTANTS. 


31 


versa.     We  must  therefore  correct  our  level  indications  by  the  effect 
of  this  "  Inequality  of  Pivots." 

To  determinate  this,  let  w  x  y  z  in  Figure  8  represent  the  cone  of 


FIG.  8. 


the  pivots,  u  v  being  the  axis.     Let  the  inclination  of  the  upper 
element  iv  z  be  measured  with  the  level,  giving 


s=(w+  w')  -  (e  +  e') 
4 


D. 


Lift  the  axis  from  the  Yys  and  turn  it  end  for  end.  In  this  position 
w'  x  y  z'  will  represent  the  cone  of  the  pivots. 

Measure  as  before  the  inclination  of  w'  z',  and  denote  it  by  B'0 
Then  by  inspection  of  the  figure  it  is  seen  that  Bf  —  B  is  the  angle 

Tlf  T) 

between  the  two  positions  of  the  upper  element,  -  -  is  the 

tit 

angle  between  the  upper  and  lower   elements  of   the   cone,  and 

T}/  7? 

—  —  p  is  consequently  the  angle  between  the  upper  element 

and  the  axis  u  v* 

*  B  and  B  are  manifestly  the  inclinations,  in  the  two  positions,  which  the 
upper  element  would  have  if  the  pivots  were  equal,  minus  twice  the  effect  of 
the  inequality: — this  effect  being  the  angle  subtended  by  the  difference  of  the 
radii,  r  —  r ' .  Of  course  if  the  pivots  are  unequal,  the  inclination  obtained  by 
applying  the  level  Y's  to  the  pivots  is  not  strictly  that  of 
the  upper  element;  but  if  the  angles  of  the  transit  and 
level  Y's  are  equal  (as  is  usually  the  case),  it  will  evidently 
be,  as  before,  the  inclination  which  the  upper  element  would 
have  if  the  pivots  were  equal,  minus  twice  the  effect  of 
the  inequality:— the  effect  in  this  case  being  (Fig.  Sa,  which 
represents  a  cross-section  of  the  pivots  and  level  T)  the 

angle  subtended  by  — — - — .     Hence    the    algebraic    differ^ 
siii  "5"  a 

ence,  B'  ~  B,  will  be  four  times  the  effect  of  the  inequality,  as  before. 


FIG.  8a. 


32  PRACTICAL  ASTRONOMY. 

T)t    T> 

This  quantity, — — —  —  p,  is  therefore  the  desired  constant,  and 

as  the  figure  indicates,  it  is  a  correction  to  be  added  algebraically 
to  the  level  determination  of  the  unreversed  instrument,  or  to  be 
subtracted  from  that  of  the  reversed  instrument. 

Its  value  should  rest  upon  many  determinations. 

The  inclination  of  the  axis  of  a  transit  will  hereafter  be  denoted 
by  b,  which  is  therefore  either  B  -j-  p,  or  B'  —  p,  according  as  the 
instrument  is  direct  or  reversed. 

»|«  The  cross-sections  of  the  pivots  should  be  perfect  circles. 
Any  departure  from  this  form  may  be  discovered  and  corrected  as 
follows : 

With  instrument  direct,  determine  the  value  of  B  with  the 
telescope  placed  successively  at  every  10°  of  altitude.  Call  the 
mean  B0 . 

Then  B0  —  B  is  the  correction  for  irregularity  of  pivots  for  the 
reading  corresponding  to  B  with  instrument  direct.  Do  the  same 
with  instrument  reversed.  Then  B0'  —  B  will  be  the  correction 

T)  r n 

for  irregularity  with  instrument  reversed.  —?-— — -  will  be  the  cor- 
rection for  inequality.  Both  corrections  must  be  applied  to  obtain 
the  true  value  of  b. 


EQUATION  OF  THE  TRANSIT  INSTRUMENT  IN  THE 
MERIDIAN. 

The  transit,  having  been  adjusted  and  the  instrumental  con- 
stants determined,  is  ready  for  use.  Hitherto  it  has  been  assumed 
that  an  adjustment  was  perfect: — that  the  middle  wire  had  been 
placed  exactly  in  the  line  of  collimation,  that  the  axis  of  rota- 
tion had  been  made  exactly  level,  and  that  the  line  of  collimation 
would  trace  with  mathematical  accuracy  the  true  meridian.  Mani- 
festly, however,  this  theoretical  accuracy  cannot  be  attained  by 
mechanical  means.  It  will  therefore  be  proper,  having  performed 
each  adjustment  as  accurately  as  possible,  not  to  regard  the  out- 
standing small  errors  as  zero,  but  to  introduce  them  into  a  given 
problem  as  additional  unknown  quantities  having  an  ascertainable 
effect  on  the  result,  and  then  to  make  independent  determinations 


TRANSIT  INSTRUMENT  IN  THE  MERIDIAN.  S3 

of  their  value,  or  leave  these  values  to  be  revealed  by  the  observa- 
tions themselves. 

Any  departure  from  perfect  adjustment  is  positive  when  its 
effect  is  to  make  stars  south  of  the  zenith  cross  the  middle  wire 
earlier  than  they  otherwise  would. 

1.  To  Ascertain  the  Effect  of  an  Error  in  Azimuth  on  the  Time 
of  Passage  of  the  Middle  Wire. — Let  a  denote  the  horizontal  angular 
deviation  of  the  axis  of  rotation  from  a  true  east  and  west  line, 
positive  when  the  west  pivot  is  south  of  the  east  pivot.  (This  should 
never  exceed  15",  and  will  usually  be  even  less.)  The  line  of  col- 
limation  will  then,  as  the  instrument  is  moved  in  altitude,  describe 
a  great  circle  of  the  celestial  sphere  intersecting  the  meridian  in 
the  zenith,  and  making  with  it  the  angle  a  (HZ  A  in  Figure  9). 


H 


FIG.  9. 


Then  from  the  Z  P  S  triangle  we  have  (S  being  the  position  of  a 
star  when  on  the  middle  wire), 

sin  P  :  sin  a  :  :  sin  z  :  cos  S, 
or 

sin  a  sin  z 
sm  P  —  -       —  =r-  - 
cos  o 

If  the  star  were  exactly  on  the  meridian,  z  would  be  equal  to 
cf>  —  d.  Being  less  than  15"  therefrom,  the  change  required  in  z 
to  give  0  —  d  is  entirely  negligible.  Again  P  and  a  are  exceed- 
ingly small  angles.  Hence  we  may  write  with  great  precision,  ex- 
pressing a  and  P  in  time, 


cos  o 


(50) 


34  PRACTICAL  ASTRONOMY. 

That  is,  if  the  instrument  have  an  azimuth  error  in  time,  of  a 
seconds,  a  star  when  passing  the  middle  wire  is  distant  from  the 

true  meridian  a 5 — -  seconds  of  time,  and  the  recorded  time 

cos  6 

of  transit  must  be  corrected  accordingly. 

2.  To  Ascertain  the  Effect  of  an  Inclination  of  the  Axis  on  the 
Time  of  Passage  of  the  Middle  Wire. — Let  b  denote  the  angular 
deviation  of  the  axis  of  rotation  from  the  horizontal,  positive  when 
the  west  pivot  is  higher  than  the  east.  The  line  of  collimation 
will  then,  as  the  instrument  is  moved  in  altitude,  describe  a  great 
circle  of  the  celestial  sphere  intersecting  the  meridian  at  the  north 
and  south  points  of  the  horizon,  and  making  with  it  the  angle  b 
(ZHS,  in  Figure  10). 


FIG.  10. 

Then  from  the  triangle  P  IIS  (8  being  the  position  of  a  star 
when  on  the  middle  wire) 

sin  P  :  sin  b  : :  cos  z  :  cos  <5. 
Or,  as  before,  expressing  b  in  time, 

p=JCO^-^)_ 

cos  d 

This  is  interpreted  as  in  the  preceding  case. 

3.  To  Ascertain  the  Effect  of  an  Error  in  Collimation  on  the  Time 
of  Passage  of  the  Middle  Wire. — Let  c  denote  the  angular  distance 
of  the  middle  wire  from  the  line  of  collimation,  positive  when  the 
wire  is  west  of  its  proper  position.  The  line  of  sight  will  then,  as 
the  instrument  is  moved  in  altitude-,  describe  a  small  circle  of  the 
celestial  sphere,  east  of  the  meridian  and  parallel  to  it.  Through 
S,  the  place  of  the  star,  Fig.  11,  pass  the  arc  of  a  great  circle,  8  M3 


TRANSIT  INSTRUMENT  IN  THE  MERIDIAN.  35 

perpendicular  to  the  meridian.     This  arc  will  be  the  measure  of 
c.     Then  in  the  right-angled  triangle  P  S  M  we  have 

.     D      sin  c 
sin  P  = 


cos 
Or,  as  before,  expressing  c  in  time, 


P  =  ~~^  =  c  sec  5.  (52) 

^     } 


cos 


Hence  when  all  these  errors,  #,  b,  and  c,  exist  together,  called  re- 
spectively the  azimuth,  level,  and  collimation  error,  we  have  for  the 
Equation  of  the  Transit  Instrument  in  the  Meridian, 


(53) 


In  this  equation  a  is  the  apparent  R.  A.  of  the  star  for  the  date, 
T  is  the  clock  time  of  transit  over  the  middle  wire,  obtained  from 
the  time  of  transit  over  the  mean  wire  by  applying  the  "  Reduction 


FIG.  11. 

to  Middle  Wire/'  E  is  the  chronometer  error,  positive  when  slow, 
negative  when  fast,  0  the  latitude,  d  the  star's  apparent  declination 
for  the  date,  and  a,  b,  and  c  are  expressed  in  time. 

When  great  precision  is  desired,  for  example  in  longitude  work, 
the  equation  must  be  modified  by  the  introduction  of  a  small  cor- 
rection for  Diurnal  Aberration,  additive  to  a.  The  value  of  the 
correction  is  Os.021  cos  0  sec  d. 

Hence  the  complete  form  of  the  above  equation  is 

a  =  T+JS  +  aS™(</'~6)   •   *«»(0-*x 


36  PRACTICAL  ASTRONOMY. 

Or,  placing 

c'  =  c  —  0.021  cos  0, 


cos  cos 

'    '  ' 


(54) 


After  an  observation  has  been  made  we  shall  have  in  this  equa- 
tion four  unknown  quantities,  E,  a,  b,  c',  since  <p  is  supposed  to  be 
known,  and  a  and  6  are  found  in  the  Ephemeris.  We  may  either 
determine  a,  b,  and  c  independently,  as  will  next  be  explained  (in 
which  case  an  observation  on  a  single  star  will  then  give  E),  or 
leave  all  four  to  be  determined  by  observation  on  at  least  four  stars. 

The  sign  of  c  is  changed  by  reversing  the  axis,  since  the  middle 
wire  is  thus  placed  on  the  other  side  of  the  line  of  collimation. 

^  This  value,  Os.021  cos  0  sec  d,  which  we  will  denote  by  R, 
may  be  deduced  in  an  elementary  manner  as  follows:  Due  to  the 
earth's  rotation  on  its  axis,  all  celestial  bodies  are  apparently  dis- 
placed toward  the  east  point  of  the  horizon.  If  the  body  be  on 
the  meridian,  this  displacement  is  wholly  in  R.  A.  Hence  the 
R.  A.  of  the  object  as  seen  will  not  be  a,  but  a  -f  R. 

The  direction  of  a  ray  of  light  received  from  a  body  on  the 
meridian  is  at  right  angles  to  the  direction  of  the  observer's  diurnal 
motion.  Under  this  condition,  the  absolute  amount  of  apparent 
displacement  in  seconds  of  a  great  circle  may  be  written  (Young, 
pa.  142), 

R  = 


Ftanl" 

where  u  is  the  observer's  velocity,  and  V  that  of  light.     If  the  ob- 
server be  at  the  equator,  we  shall  have 

20926062  X  2  n 
~-  "5280  x  24  X  60  >^0  mileS  per  SeC°nd> 

where  20926062  is  the  number  of  feet  in  the  earth's  equatorial 
radius  (Clarke). 

According  to  Newcomb  and  Michelson, 

V  =  186330  miles  per  second. 


DETERMINATION  OF  INSTRUMENTAL  ERRORS.  3? 

Hence 

R  =  _  20926062  x  2  n  _ 

5280  X  24  X  3600  x.  186330  X  tan  1"  " 

This  angular  displacement  in  a  great  circle  perpendicular  to  the 
meridian  corresponds  to  09.021  if  the  star  be  on  the  equator,  or  to 
0?021  sec  d  if  the  star's  declination  be  #,  since,,  as  we  have  seen 
before,  equal  angular  distances  from  the  meridian  correspond  to 
hour  angles  varying  with  sec  ft. 

If  the  observer  be  not  on  the  equator,  but  at  latitude  <f),  his 
velocity  will  be  diminished  in  the  ratio  of  the  radius  of  his  circle  of 
latitude  to  that  of  the  equator:  or  regarding  the  earth  as  a  sphere, 
in  the  ratio  cos  4>  :  1. 

Hence,  for  an  observer  in  any  latitude,  with  a  star  at  any  dec- 
lination, 

R  =  Os.021  cos  0  sec  d. 

DETERMINATION   OF   INSTRUMENTAL   ERRORS. 

1.  To  Determine  the  Level  Error  b.  —  This  is  found  from  the 
formula  already  deduced,  viz.  : 


D  +p          (55) 

or 
:;.;      .       V  =  B'-f  =  ^  +  ^-^  +  ^D-f,  (56) 

according  as  the  instrument  is  direct  or  reversed.  D  and  p  must 
be  expressed  in  time,  by  dividing  their  values  in  arc  by  15,  thus 
giving  b  in  time. 

2.  To  Determine  the  Collimation  Error  c.  —  Turn  the  instrument 
to  the  horizon,  select  some  well-defined  distant  point  whose  image 
is  near  the  middle  wire,  measure  the  distance  between  them  with 
the  micrometer,  making  the  distance  positive  when  the  middle 
wire  is  west  of  the  image  of  the  point.  Reverse  the  axis,  and  meas- 
ure the  new  distance,  with  same  rule  as  to  sign.  Subtract  the 
second  from  the  first,  and  one  half  the  difference  gives  the  colli- 
mation  error  in  micrometer  divisions  for  instrument  direct. 


38  PRACTICAL  ASTKONOMY. 

This  multiplied  by  the  value  of  one  division  in  time,  gives  c 
in  time. 

The  rule  will  be  evident  from  an  inspection  of  Fig.  12  (which 
is  a  horizontal  projection),  where  w  is  the  west, 
and  e  the  east  end  of  the  axis,  T  E  the  hori- 
zontal line  of  collimation,  P  the  image  of  the 
e    point  in  the  field  of  view,  a  the  direct  and  b 
the  reversed  position  of  the  middle  wire.     E  a 
>_^_          is  equal  to  E  b,  and  c  is  positive. 

b  Instead  of  a  terrestrial  point  we  may  use  the 

FIQ.  12.  intersection  of  the  cross  hairs  in  the  focus  of 

a  surveyor's  transit  adjusted  to  stellar  focus, 
the  two  instruments  facing  each  other.  The  intersection  referred 
to  will  then  be  optically  at  an  infinite  distance,  and  its  image  will 
be  found  at  the  principal  focus  of  our  transit. 

It  is  sometimes  necessary  to  determine  c  by  independent  stellar 
observations,  in  which  case  the  following  method  is  always  employed  : 
Point  the  telescope  to  a  circumpolar  star  and  note  the  times  of  its 
passage  over  as  many  wires  as  possible  on  one  side  of  the  middle 
wire.  Eeverse  the  axis.  As  the  star  moves  out  of  the  field  of  view, 
it  will  cross  the  same  wires  in  reverse  order,  the  times  of  passage 
being  noted  as  before. 

By  means  of  the  Equatorial  Intervals  reduce  each  time  to  the 
middle  wire,  and  let  T  and  T'  denote  the  mean  of  those  before  and 
after  reversal,  respectively. 

T  and  Tf  are  therefore  the  times  of  passage  of  the  same  star 
over  two  different  positions  of  the  middle  wire  —  one  as  much  to 
the  east  as  the  other  was  to  the  west  of  the  true  line  of  collimation. 
From  their  difference  therefore  we  have  double  the  collimation  error, 
thus: 

For  instrument  direct, 


sin 


T_I_  w  _L  -       ,  - 

a  —  1  4-  E  4-  a  --  5  —  *  4-  0  —       —  ^  -- 


cos(0-tf)         c          Os.021  cos  0 


5  —     -     —       —  ^  --  ---  j  --     —  -T 
cos  o  cos  o  cos  o  cos  o 

For  instrument  reversed, 

„,   .    v  ,   >n(0-tf)   ,   ,,,008(0-0)          c         0".021*) 
Ct  ^^  -/    •+•  Hi  ~T~  u  —  —  -  jc  -  ~i    "  ~~         —  ?i  —  _n  "F~ 

COS  O  COS  O  COSO  COk  C 


DETERMINATION  OF  INSTRUMENTAL  ERRORS.  39 

allowance  being  made  for  a  change  in  level  error  due  to  a  possible 
inequality  of  pivots,  and  c  changing  its  sign  by  reversal  of  the  in- 
strument. 

By  subtraction  and  solution  we  have 


c  =  i  (r  -  T)  cos  tf  +  i  (V  -  b)  cos  (0  -  tf).          (57) 

If  the  pivots  are  equal  and  the  instrument  be  undisturbed  in 
level,  the  last  term  disappears  and  we  have 

c  =  %(T'  -  T)  cos  d.  (58) 

A  slow-moving  star  must  be  used  in  order  to  give  time  for  care- 
ful reversal. 

There  are  various  other  methods  of  finding  both  b  and  c,  based 
principally  upon  observation  of  the  wires  and  their  images  as  seen 
by  reflection  from  mercury. 

3.  To  Determine  the  Azimuth  Error,  a.—  Observe  in  the  usual 
manner  the  time  of  transit,  T,  of  a  star  of  known  declination. 
Then,  b  and  c  having  been  measured,  let  the  corresponding  correc- 

tions, b  —  •-  —  -r  -  and  cf  sec  d,  be  added  to  T,  giving  t.     This  is 
cos  o 

called  correcting  the  time  for  level  and  collimation.     The  equation 
of  the  instrument  as  applied  to  this  star  will  now  read 

sin  (0  —  d} 

a=^t  +  E  +  a  -  —  -~^—L.  (m) 

cos  d 

Similarly  for  another  star, 

a>=t>  +  E+aS™(<t>-S'l  (n) 

cos  6'  v  ' 

From  which 


a  (sin  0  —  cos  0  tan  df  —  sin  0.-f  cos  0  tan  #)  =  («'  —  a)  —  (tf—  t). 

(a>  -a)-  (('  -  t) 
'  cos  0  (tan  $  -  tan  <J')'  l    ' 


40  PRACTICAL  ASTRONOMY. 

The  value  of  the  clock  error  does  not  enter.  If  however  it  be 
not  constant,  its  rate,  r,  must  be  known,  positive  when  losing,  nega- 
tive when  gaining.  Then  if  EQ  be  the  unknown  error  at  some  as- 
sumed instant  T0 ,  the  errors  at  the  two  instants  of  observation  will 
be  J£0+  (T-T,)  r,  and  £0+  (T'  -  T9)  r.  These  should  be  substi- 
tuted for^in  equations  (m)  a-nd  (n),  and  the  known  terms,  ( T—  T0)  r 
and  (Tf—  T0)  r,  be  united  to  T  and  T'  in  forming  t  and  t'  as  are  the 
corrections  for  level  and  collimation.  The  time  is  then  said  to  be 
corrected  for  rate.  By  subtraction  to  obtain  (59),  E0  will  disappear. 
Hence  while  the  rate  must  be  known,  the  error  need  not  be. 

Examining  the  value  of  a,  we  see  that  the  following  conditions 
must  be  fulfilled  in  order  to  obtain  an  accurate  determination. 

First,  a  and  a'  must  be  known  exactly;  therefore  only  Ephem- 
eris  stars  should  be  used. 

Again,  if  the  rate  of  the  clock  be  not  well  determined,  the 
interval  between  the  observations  must  be  as  small  as  possible  in 
order  that  the  correction  for  rate  may  affect  a  but  slightly.  There- 
fore if  both  stars  are  at  upper  culmination,  they  should  be  nearly 
equal  in  R.  A.  Or,  if  one  be  above  and  the  other  below  the  pole, 
they  should  differ  in  R.  A.  by  as  nearly  12  hours  as  possible. 

Again  the  larger  numerically  the  factor  (tan  d  —  tan  d'),  the  less 
the  effect  of  errors  in  tf  —  t.  Hence,  if  both  stars  are  at  upper 
culmination,  one  should  be  as  near  and  the  other  as  far  from  the 
pole  as  possible.  Or,  if  one  be  at  upper  and  one  at  lower  culmina- 
tion, they  should  both  be  as  near  the  pole  as  possible;  the  declina- 
tion of  the  lower  star  being  then  taken  to  be  90°  -f-  Polar  Distance. 


Erom  the  preceding  description  of  the  Transit  Instrument  it 
be  readily  understood,  that,  if  desired,  the  mean  wire  may  be 
used  as  a  datum  instead  of  the  middle,  and  the  Equatorial  Intervals 
be  determined  from  it  with  the  same  facility  as  from  the  middle 
wire.  Also,  if  in  Eq.  (53)  T  be  the  time  of  a  star's  transit  over 
the  mean  wire,  c  will  be  the  collimation  error  of  this  wire,  and, 
together  with  E,  a,  and  Z>,  may  be  determined  by  the  use  of  four 
stars  as  explained  on  page  36.  It  may  also  be  determined  by  Eq. 
(57)  or  (58),  if  T  and  T'  be  computed  for  the  mean  instead  of  the 
middle  wire.  This  use  of  the  mean  for  the  middle  wire  is  frequent 
in  field  work,  and  possesses  the  advantage  that  all  consideration  of 
the  "Reduction  to. the  Middle  Wire"  may  be  then  avoided. 


REFRACTION  TABLES.  41 

REFRACTION  TABLES. 

A  ray  of  light  passing  from  a  celestial  body  to  a  point  on  the 
earth's  surface,  may  be  supposed  to  pass  through  successive  spherical 
strata  of  the  atmosphere,  the  densities  of  which  continually  increase 
•toward  the  center.  Under  these  circumstances,  as  has  been  previ- 
Dusly  shown,  the  ray  will  be  bent  toward  the  normal,  resulting  in 
an  apparent  displacement  of  the  body  toward  the  zenith. 

It  has  also  been  previously  shown  that  the  actual  amount  of  such 
displacement  increases  with  the  zenith  distance,  and  with  the 
density  of  the  air,  which  latter  depends  on  its  pressure  and  tempera- 
ture. In  order  to  facilitate  the  calculation  of  this  displacement  or 
refraction  in  any  particular  case,  tables  have  been  constructed  con- 
containing  certain  functions  of  the  zenith  distance,  temperature, 
and  pressure,  from  which,  with  observed  data  as  arguments,  the  re- 
fraction may  be  computed. 

Such  tables  are  called  Refraction  Tables.  Those  of  Bessel  are 
the  best  and  most  usually  employed.  In  these  tables  the  adopted 
value  of  the  refraction  function  is  given  by 

r  =  a  j3  yx  tan  z, 

in  which  r  is  the  refraction;  A,  A,  and  a  are  quantities  varying 
slowly  with  the  zenith  distance;  /3  is  a  factor  depending  on  the 
pressure,  and  y  upon  the  temperature  of  the  air;  z  is  the  apparent 
zenith  distance  ;  ft  therefore  depends  upon  the  reading  of  the  ba- 
rometer, and  y  upon  the  reading  of  the  thermometer.  But  since 
the  actual  height  indicated  by  a  barometer  depends  not  only  upon 
the  pressure  of  the  air,  but  upon  the  temperature  of  the  mercury, 
/3  is  really  composed  of  two  factors  B  and  T,  the  first  of  which  de- 
pends upon  the  actual  reading  of  the  barometer,  and  T  involves  the 
correction  due  to  the  temperature  of  the  mercury. 

Nearly  all  the  collections  of  astronomical  tables  contain  "  Tables 
of  Refraction,"  from  which  may  be  found  the  various  quantities  in 
the  equation 


The  first  portion  of  the  table  consists  of  three  columns  giving  the 
values  of  A,  A,  and  log  a,  with  the  apparent  zenith  distance  z  as 
the  argument. 


2  PRACTICAL  ASTRONOMY. 

The  second  part  contains  B,  with  the  height  of  the  barometer 
as  the  argument.  The  third  part  gives  the  value  of  T  with  the 
reading  of  the  attached  thermometer  as  the  argument,  and  the 
fourth  part  gives  y  with  the  reading  of  the  external  thermometer 
as  the  argument;  z  is  the  observed  zenith  distance.  A  substitution 
of  these  quantities  gives  the  refraction,  which  must  then  be  added 
to  z  to  give  the  true  zenith  distance. 

The  attached  thermometer  gives  the  temperature  of  the  mercury 
of  the  barometer.  The  external  thermometer  should  be  screened 
from  the  direct  and  reflected  heat  of  the  sun,  but  be  so  fully  ex- 
posed as  to  give  accurately  the  temperature  of  the  external  air. 

A  similar  table  is  sometimes  given  for  passing  from  true  to  ap- 
parent zenith  distances.  The  mode  of  using  is  exactly  the  same, 
subtracting  the  resulting  refraction  from  the  true  zenith  distance 
to  obtain  z.  It  is  of  use  in  "setting"  instruments  for  observation. 

A  "  Table  of  Mean  Refractions "  is  also  given  in  nearly  every 
collection,  and  contains  the  refractions  for  a  temperature  of  50°  F., 
and  30  in.  height  of  barometer,  with  apparent  zenith  distances  or 
altitudes,  as  the  argument,  which  may  be  used  when  a  very  precise 
result  is  not  required. 

The  above  relates  only  to  refraction  in  altitude.  But  a  change 
in  a  star's  place  due  to  refraction  will  in  the  general  case  cause  a 
change  in  its  observed  R.  A.  and  Dec.  In  order  to  ascertain  these 
two  coordinates  as  affected  by  refraction  at  a  given  sidereal  time  T, 
we  first  compute  the  body's  hour  angle  from  P  =  T  —  R.  A.,  and 
then  its  true  zenith  distance  (z)  and  parallactic  angle  (^)  from  the 
astronomical  triangle,  knowing  P,  (p,  and  tf.  Then  if  r  denote  the 
refraction  in  altitude,  found  as  just  explained,  the  refraction  in 
declination  will  be 

A  $  —  r  cos  if?, 
and  the  refraction  in  R.  A., 

r  sin  ib 

A  a  = X.     • 

cos  # 

TIME. 

The  perfect  uniformity  with  which  the  earth  rotates  on  its  axis 
makes  its  motion  a  standard  regulator  for  all  time-pieces.  No  clock 
or  chronometer  can  run  with  perfect  uniformity,  an^  therefore  the 
time  indicated  by  them  must  ever  be  in  error.  To  find  these  errors 
at  any  instant  is  the  object  of  the  time  problems  in  Practical  As- 
tronomy. 


TIME.  43 

Time  is  measured  by  the  hour  angle  of  some  point  or  celestial 
body.  If  the  point  be  the  true  Vernal  Equinox  its  hour  angle  is 
true  sidereal  time. 

If  the  point  be  the  mean  Equinox,  it  is  mean  sidereal  time;  but 
since  the  greatest  difference  between  true  and  mean  sidereal  time 
can  never  exceed  1.15  seconds  in  19  years,  astronomical  clocks  are 
run  on  true  sidereal  time.  To  pass  from  true  to  mean  sidereal 
time,  apply  the  correction  known  as  the  Equation  of  Equinoxes  in 
Eight  Ascension. 

If  the  point  be  the  Mean  Sun  its  hour  angle  is  mean  solar  time; 
all  solar  time  pieces  are  run  on  mean  solar  time. 

If  the  point  be  the  center  of  the  True  Sun,  its  hour  angle  is 
true  or  apparent  solar  time;  to  pass  from  true  to  mean  solar  time 
apply  the  correction  known  as  the  Equation  of  Time. 

Before  proceeding  to  the  time  problems,  it  is  necessary  to  deter- 
mine the  relation  existing  between  sidereal  and  mean  solar  intervals, 
and  especially  the  relation  existing  between  the  sidereal  and  mean 
solar  time  at  any  instant. 

Relation  between  Sidereal  and  Mean  Solar  Intervals.  —  The  in- 
terval of  time  between  two  consecutive  returns  of  the  sun  to  the 
mean  vernal  equinox,  called  the  mean  tropical  year,  is  according  to 
Bessel  365.2422  mean  solar  days.  Since,  while  the  earth  is  rotating 
on  its  axis  from  west  to  east,  the  mean  sun  is  moving  uniformly  in 
the  same  direction,  the  interval  between  two  consecutive  passages 

of  the  meridian  over  the  mean  sun  will  be  1  +  times  the 


interval  between  two  passages  .  over  the  mean  vernal  equinox:  for 
in  one  mean  solar  day  the  mean  sun  must  advance  •    of  the 


whole  circuit  from  equinox  to  equinox,  and  each  mean  solar,  day 
must  correspond  to  1  -}-—.-  0.<oo  passages  of  the  mean  vernal  equi- 


nox.     Hence   365.2422   mean  solar  days    correspond  to   366.2422 
sidereal  days. 

Hence  we  have  the  relations, 

366  9422 
One  mean  solar  day  =  365]2422  =  1-00273791  sidereal  days, 

=  24h  3m  5  6  ".555  sidereal  time. 


44  PRACTICAL  ASTRONOMY. 

365  ^4^9 
One  sidereal  day  =        '^  -^  =  0.99726957  metm  so/ar  days, 


=  23h  56m  4S.091  mean  solar  time. 

The  same  relation  manifestly  exists  between  the  corresponding 
hours,  minutes,  and  seconds..  Now  since  the  sidereal  unit  is  shorter 
than  the  mean  solar  in  the  ratio  of  1  :  1.00273791,  it  follows  that 
the  number  of  these  units  in  a  given  interval  of  time  is  to  the 
number  of  mean  solar  units  as  1.00273791  to  1. 

HeiLce  the  relations, 

Sidereal  Interval  —  Mean  Solar  Interval  X  1.00273791. 
Mean  Solar  Interval  =  Sidereal  Interval  X  0.99726957. 

Or,  denoting  these  intervals  respectively  by  F  and  /, 

F  =  /+  0.00273791  / 
/  =  /'—  0.00273043  F, 

Tables  II  and  III,  Appendix  to  the  Ephemeris,  give  the  values  of 
the  corrections  O.Q0273791  /and  0.00273043  /',  for  each  second  in 
the  24  hours. 

Again,  since  24  sidereal  hours  equals  23h  56ta  4S.091  mean  solar 
time,  it  follows  that  a  mean  solar  clock  loses  3m  55S.909  on  a  side- 
real clock  in  one  sidereal  day,  or  98.8296  in  one  sidereal  hour. 

Also,  since  24  mean  solar  hours  equals  24h  3m  568.555  sidereal 
time,  it  follows  that  a  sidereal  clock  gains  3m  56S.555  on  a  mean 
solar  clock  in  one  mean  solar  day,  or  98.8565  in  one  mean  solar 
hour. 

These  two  facts  may  be  thus  expressed  : 

(1)  The  hourly  rate  of  a  mean  solar  clock  on  sidereal  time  is 
+  99.8296. 

(2)  The  hourly  rate  of  a  sidereal   clock  on  mean   solar   time   is 
-  9S.8565. 

From  (2)  it  is  seen  that  the  E.  A.  of  the  mean  sun  increases 
98.8565  per  hour,  or  in  other  words,  the  sidereal  time  of  mean  noon 
occurs  98.8565  later  for  each  hour  of  west  longitude. 

These  deductions  are  of  importance  in  what  follows. 

Relation  between  Sidereal  and  Mean  Solar  Time.  That  is,  haying 
given  either  the  sidereal  or  mean  solar  time  at  a  certain  instant,  to 
End  the  other. 


TIME.  40 

Suppose  first  the  sidereal  time  to  be  given  and  let  the  circle  in 
Figure  13  represent  the  celestial 
equator,  M  being  the  point  where  it 
is  intersected  by  the  meridian,  V  the 
vernal  equinox  and  S  the  place  of 
the  mean  sun. 

Then  MV=  the  sidereal  time  at 
the  instant,  supposed  to  be  known, 
MS  =  the  mean  solar  time  required, 
and  V  8  =  right  ascension  of  mean 
sun. 

The  mean  solar  time  required  is 
therefore  equal  to  the  given  sidereal  FlG.  ]3. 

time  minus  the  R.  A.  of  the  mean  sun  at  the  instant.  The  calcu- 
lation of  the  R.  A.  of  the  mean  sun  at  a  given  instant  may  be 
avoided  by  the  use  of  Tables  II  and  III,  Appendix  to  the  Ephem- 
eris,  as  follows : 

At  the  preceding  mean  noon  the  mean  sun's  R.  A.  was  less  than 
at  the  moment  considered  by  an  amount  which  may  be  represented 
ly  ##'. 

At  that  time,  therefore,  the  mean  sun  was  at  M,  and  the  Vernal 
Equinox  at  a  position  V  such  that  V  M  =  VS'.  Hence  at  the 
instant  considered,  the  sidereal  time  elapsed  since  the  preceding 
mean  noon  is  MV  —  MV.  The  time  since  mean  noon  having  thus 
been  found  in  sidereal  units,  the  mean  solar  equivalent  of  this  inter- 
val will  necessarily  be  the  mean  solar  time  at  the  instant  consid- 
ered. Hence  the  rule : 

From  the  given  sidereal  time  subtract  the  R.  A.  of  the  mean  sun 
at  the  preceding  mean  noon.  Convert  the  result  into  a  mean  solar 
interval  by  theEphemeris  Tables  or  theformulaI=F— 0.002730437'. 
The  result  is  the  required  mean  solar  time. 

To  find  the  sidereal  from  the  given  mean  time,  this  operation 
must  obviously  be  performed  in  the  inverse  order,  viz. : 

Convert  the  given  mean  solar  time  into  a  sidereal  interval  by 
the  Ephemeris  Tables  or  by  the  formula  I*  =  1+  0.00273791/. 
To  the  result  add  the  R.  A.  of  the  mean  sun  at  the  preceding  mean 
noon.  The  result  is  the  required  sidereal  time. 

On  Page  II,  Monthly  Calendar  of  the  Ephemeris,  will  be  found 
the  R.  A.  of  the  mean  sun  at  the  preceding  Greemvich  mean  moon. 
To  find  this  element  for  the  local  mean  noon,  multiply  the  hourly 


46  PRACTICAL  ASTRONOMY. 

change  98.8565  (heretofore  deduced)  by  the  longitude  in  hours,  and 
add  the  result  to  the  Ephemeris  value. 

The  above  rules  are  not  only  of  great  use  in  astronomical  calcu- 
lations, but  they  enable  us  to  determine  the  error  of  either  a  side- 
real or  mean  time  clock,  knowing  that  of  the  other,  by  "the  method 
'of  coincident  beats."  Suppose  both  clocks  to  beat  seconds.  Then 
from  the  relative  rate  heretofore  deduced  it  is  seen  that  their  beats 
will  be  coincident  once  in  about  6  minutes.  Note  the  seconds  given 
by  each  clock  when  this  occurs,  and  then  supply  the  hours  and 
minutes.  Apply  the  known  error  to  the  mean  solar  for  example ; 
and  the  result  will  be  the  correct  m.  s.  time.  Find  the  correspond- 
ing sidereal  time  by  the  rule  just  given.  The  difference  between 
this  and  the  time  given  by  the  sidereal  clock  will  be  its  error. 

EXAMPLE. 

At  West  Point,  N.  Y.,  Nov.  27,  1891,  Longitude  4h.93  west,  the 
mean  solar  and  sidereal  clocks  were  compared  at  the  instant  of 
coincident  beats,  with  the  following  result : 

Mean  Solar,   Oh  -  46m-  298.00. 
Sidereal,       17h  -  15m-  55s.OO. 

The  error  of  the  mean  solar  was  OM7  slow  on  Standard  Time, 
which  is  itself  4m  98.45  slow  on  local  time. 

It  is  required  to  find  the  error  of  the  sidereal  clock. 

Indicated  m.  s.  time Oh  —  46m—  29".00 

Error  on  standard  time 0.17 

Reduction  to  local  time 4    —    9.45 

Corrected  m.  s.  time 0    —  50  —  38.62 

Reduction  to  sidereal  interval 8.32 

Sidereal  interval  since  mean  noon 0    —  50  —  46.94 

R.  A.  of  mean  sun  at  Greenwich  mean  noon 16    —  24  —  27.25 

Correction  =  98.8565  X  4.93 48.59 

True  sidereal  time 17   —  16  -      2.78 

Clock  indication 17    —   15  —  55.00 

Error  of  sidereal  clock -|-  7,78 

Hence  the  sidereal  clock  was  7'.78  slow. 


TO  FIND  THE  TIMS  BY  ASTRONOMICAL  OBSERVATIONS.     47 


TO  FIND  THE  TIME  BY  ASTRONOMICAL  OBSERVA- 
TIONS. 

This  general  problem  usually  presents  itself  as  a  question  of  de- 
termining the  error  of  a  time-piece  at  a  given  instant.  The  different 
methods  of  obtaining  this  error  may,  as  far  as  considered  here,  be 
grouped  under  three  heads. 

/.  Time  by  Meridian  Transits. 
II.   Time  by  Single  Altitudes. 

III.   Time  by  Equal  Altitudes. 

The  first  is  the  method  of  precision  when  properly  carried  out 
with  the  transit  instrument.  The  second  and  third,  being  usually 
carried  out  with  the  sextant,  can  only  be  relied  upon  as  giving  an 
approximate  result  more  or  less  exact. 

I.    TIME    BY    MERIDIAN   TRANSITS. 

1.  To  Find  the  Error  of  a  Sidereal  Time-piece  by  the  Meridian 
Transit  of  a  Star.  (See  Form  1.) — The  general  statement  of  the 
problem  is  briefly  this:  since  the  time-piece,  if  correct,  ought  to  in- 
dicate the  R.  A.  of  the  star  at  the  instant  of  culmination,  the  dif- 
ference in  time  is  the  error  required.  The  transit  instrument  being 
supposed  to  be  approximately  in  the  meridian,  i.e.,  to  have  been 
carefully  adjusted,  for  the  practical  solution  it  is  necessary  to  find 
by  observation  and  computation  the  quantities  in  the  following 
equation  (heretofore  deduced)  and  solve  it. 

a  -  T+  E  +  aA  +  bB  +  c'C,  (60) 

in   which   A,   B,  and    C  have  for   brevity   been   substituted   for 

sin  (0  —  6}      cos  (0  —  tf)        ,          ~  ,.     ,        mi       , 

i2- — - — '-,        — 5 — -,  and  sec  tf,  respectively.     Then  having 

cos  d  cos  d 

measured  a,  b,  and  c;  computed  A,  B,  and  C\  observed  T7;  and 
taken  a  from  the  Ephemeris  (a  =  the  star's  apparent  R.  A.  for  the 
date),  the  value  of  E  follows  from  the  solution  of  ths  equation. 

In  finding  T7,  record  to  quarter  seconds  (or  if  possible  to.  tenths 
of  a  second),  the  time  of  passage  of  each  wire.  Take  the  mean 
and  apply  the  "Reduction  to  middle  wire."  T,  corrected  by 
a  A  -f-  bB  +  c'C  is  evidently  the  chronometer  time  of  the  star's 
transit  over  the  meridian. 


48  PRACTICAL  ASTRONOMY. 

Form  1  indicates  the  proper  method  of  recording  the  observa- 
tions, it  being  arranged  for  five  stars.  Under  the  head  of  "  Transit/7 
record  its  number  and  the  maker.  The"  Illumination"  should  be 
recorded  as  east  or  west,  this  showing  whether  the  instrument  is 
direct  or  reversed. 

The  adopted  value  of  'E  should  &e  the  mean  of  the  results  from 
several  stars.  Stars  within  the  polar  circle,  or  those  whose  declina- 
tion exceeds  about  67°,  are  not  used  for  time  determinations,  since 
the  exact  instant  when  a  slow  moving  star  is  bisected  by  a  wire  can- 
not be  judged  with  the  greatest  precision,  and  since  also  slight 
errors  in  measuring  a,  b,  and  c  will  then  be  greatly  magnified  by  A, 
B,  and  C,  all  of  which  become  oo  for  6  —  90°.  But  by  including 
in  the  observing  list  two  circum-polar  stars  upon  one  of  which  the 
instrument  is  reversed  after  half  the  wires  are  passed,  both  a  and  c 
may  be  found  by  Equations  (57)  and  (59).  b  is  found  from  level 
readings  by  Equation  (55)  or  (56). 

If  only  a  single  star  is  available,  it  should  be  one  given  in  the 
Ephemeris,  and  which  passes  near  the  zenith  (d  =  0),  since  at  the 
zenith  Aa  disappears,  and  this  is  the  only  one  of  the  three  correc- 
tions which  requires  star  observations  for  its  determination. 

For  very  accurate  work,  such  as  is  required  in  connection  with 
the  telegraphic  determination  of  longitude,  it  is  usual  to  employ  at 
least  ten  stars  for  each  determination  of  time,  half  the  stars  being 
observed  with  the  instrument  reversed;  and  of  each  half,  two  should 
be  circum-polar  and  three  equatorial  stars.  In  this  case,  b  is  ordi- 
narily the  only  instrumental  error  actually  measured  (by  level  read- 
ings) ;  each  star  then  gives  an  equation  of  the  form  (60),  and  E  to- 
gether with  a  and  c  are  found  from  a  solution  of  the  equations  by 
Least  Squares. 

These 'matters  will  be  explained  more  fully  hereafter. 

If  the  "Reduction  to  the  Middle  Wire"  be  not  applied  in  com- 
puting T,  c  will  be  the  collimation  error  of  the  mean  wire.  This 
fact  is  of  general  application  whenever  the  Transit  Instrument  is 
used  for  determining  Time. 

The  clock  rate  is  found  from  errors  determined  at  different 
times. 

To  find  the  error  at  a  given  instant,  as  for  example  at  the  middle 
of  the  time  consumed  in  a  series  of  observations  extending  over 
several  hours,  this  rate  should  be  applied  as  explained  when  treating 
of  the  azimuth  error. 


TO  FIND  THE  TIME  BY  ASTRONOMICAL  OBSERVATIONS.   49 

*J*  It  may  sometimes  be  desirable  to  find  the  error  of  a  sidereal 
clock  from  a  meridian  transit  of  the  sun,  although  in  field  work 
this  would  be  exceptional.  In  such  a  case  it  may  be  assumed,  with 
an  error  entirely  negligible,  that  during  the  short  time  consumed  in 
the  observation  the  sun's  motion  is  uniform,  that  the  time  required 
for  the  sun  to  pass  from  the  mean  to  the  middle  wire,  and  from  the 
middle  wire  to  the  meridian  is  the  same  as  that  for  a  star  of  the 
same  declination. 

For  example,  the  reduction  to  middle  wire  not  exceeding  08.5, 
the  error  committed  by  the  second  assumption  could  not  exceed 
AS  00974 
— ~  *  X  sec  (23°  28')  =  Os.0015.      Hence  that  reduction  may  be 

computed  as  usual. 

Therefore,  note  the  time  of  transit  of  each  limb  of  the  sun  over 
each  wire,  and  take  the  mean.  Eeduce  to  the  middle  wire  as  usual, 
and  apply  the  correction  a  A  -}-  bB  +  c'C.  The  result  is  the  clock 
time  of  culmination  of  the  sun's  center.  The  true  sidereal  time  of 
this  event,  or  the  E.  A.  of  the  sun  at  apparent  noon,  is  found  on  page 
1,  Monthly  Calendar,  by  interpolation.  The  difference  gives  E. 


2.  To  Find  the  Error  of  a  Mean  Solar  Time-piece  by  a  Meridian 
Transit  of  the  Sun.  (See  Form  2.) — Apparent  noon  at  any  place  is 
the  instant  of  culmination  of  the  sun's  center  at  that  place.  This 
epoch  may  be  expressed  in  three  different  times,  viz. : 

In  apparent  time,  or  12  o'clock  apparent  time. 

In  mean  time,  or  12  o'clock  plus  the  equation  of  time. 

In  clock  time,  or  that  indicated  by  a  mean  solar  time-piece. 

At  apparent  noon  a  mean  solar  time-piece  should  therefore  indi- 
cate 12  o'clock  plus  the  equation  of  time  at  the  instant. 

Therefore  the  general  equation  of  the  Transit  Instrument  be- 
comes for  this  case 

12h  +  e  =  T+E+aA+bB  +  c'C,  (61) 

e  denoting  the  equation  of  time. 

Note  the  order  and  directions  that  follow : 

1.  The  mean  of  all  the  observed  times  is  the  chronometer  time  of 

transit  of  sun's  center  over  the  mean  of  the  wires. 

2.  The  reduction  to  middle  wire,  as  well  as  the  three 'corrections, 

are  found  as  in  Form  1.     By  adding  them  to  the  above-men- 


50  PRACTI6AL  ASTRONOMY. 

tioned  mean,  we  have  the  chronometer  time  of  apparent  noon. 
The  declination  of  the  sun,  used  in  computing  these  corrections, 
is  to  be  taken  from  the  Ephemeris,  allowance  being  made  for 
the  observer's  longitude.  Use  page  1,  Monthly  Calendar. 

3.  The  mean  time  of  apparent  noon  is  12  hours  -f  e.  In  comput- 
ing e  use  page  1,  Monthly  Calendar,  and  make  allowance  for 
observer's  longitude.  The  Ephemeris  gives  the  sign  of  e. 

&  Subtract  the  chronometer  time  of  apparent  noon  from  the  mean 
time  of  apparent  noon,  and  the  remainder  is  the  error  of  the 
chronometer: — plus  if  slow,  minus  if  fast. 

5.  Time-pieces  at  West  Point  are  run  on  75th  Meridian  mean  time, 
i.e.  4I1X  9S.45  slower  than  local  mean  time.  Hence  in  finding 
the  error  at  West  Point  subtract  4m  9S.45  from  12h  ±  e,  before 
proceeding  with  step  No.  4. 


•%*  Should  the  necessity  arise  for  finding  the  error  of  a  mean 
solar  time-piece  by  a  meridian  transit  of  a^star,  it  may  b*e  done  by 
the  same  methods,  the  reduction  to  the  middle  wire  and  corrections 
for  instrumental  errors  being  computed  as  usual,  since  the  equa- 
torial value  of  the  first,  as  before,  being  taken  as  not  exceeding  08.5, 

AS  00273 

the  greatest  error  thus  produced  cannot  exceed   -  -  sec  67° 

Z 

—  0s. 0035.     Stars  within  the  polar  circle,  or  whose  declination  ex- 
ceeds about  67°  are  not  used  for  time  determinations. 

Therefore  having  observed  the  clock  time  of  transit  (corrected  by 
aA  +  bB-\-cC),  and  having  computed,  as  heretofore  explained, 
the  correct  mean  solar  time  of  transit  from  the  star's  E.  A.,  the  dif- 
ference gives  the  clock  error. 


THE   SEXTANT. 

As  problems  under  the  second  and  third  heads  arising  MI  field 
work,  are  usually  solved  by  aid  of  the  sextant,  a  short  description  of 
that  instrument  and  the  manner  of  using  it  becomes  necessary. 

The  sextant  is  a  hand  reflecting  instrument  designed  for  the 
measurement  of  the  angular  distance  between  two  objects.  In  its 
construction  it  embodies  the  following  principle  of  Optics,  viz.s 


TO  FIND  THE  TIME  BY  ASTRONOMICAL  OBSERVATIONS.     51 

When  a  ray  of  light  is  reflected  successively  by  two  plane  mirrors, 
the  angle  between  the  first  and  last  direction  of  the  ray  is  twice  the 
angle  between  the  mirrors,  provided  the  ray  and  its  two  reflections 
are  all  in  the  same  plane  perpendicular  to  both  mirrors.  For  as- 
tronomical work  the  sextant  is  mainly  used  for  measuring  vertical 
angles,  i.e.,  the  altitude  of  some  celestial  body.  In  the  measure- 
ment of  Lunar  Distances,  however,  the  angle  will  usually  be  in- 
clined. 

The  instrument  consists  essentially  of  a  graduated  circular  arc, 
usually  somewhat  over  90°  in  extent,  connected  with  its  center  by 
several  radii  and  braced  by  cross  pieces,  forming  what  is  known  as 
the  frame.  Attached  to  the  center  of  the  arc  is  a  movable  index- 
arm  provided  with  clamp  and  tangent  screw,  carrying  at  its  outer 
end  a  vernier  and  microscope  for  reading  the  sextant  arc.  Attached 
to  the  index-arm  at  its  center  of  motion,  and  therefore  rotating  with 
it,  is  a  small  mirror  known  as  the  index-glass,  whose  plane  is  per- 
pendicular to  that  of  the  frame.  Perpendicular  to  the  frame,  at- 
tached thereto  and  therefore  immovable,  is  a  second  small  mirror, 
known  as  the  horizon- glass.  These  two  mirrors  are  so  placed  with 
reference  to  each  other  that  when  the  index-arm  vernier  points  to 
the  zero  of  the  arc,  they  shall  be  exactly  parallel  and  facing  each 
other.  In  this  position  a  ray  reflected  by  both  mirrors  will  have  its 
original  direction  unchanged.  The  horizon-glass  is  divided  into 
two  parts  by  a  line  parallel  to  the  frame.  The  first  part  next  the 
frame  is  a  mirror,  and  is  the  horizon-glass  proper.  The  outer 
part,  consisting  of  unsilvered  glass,  is  not  a  mirror.  A  small  tele- 
scope screwing  into  a  fixed  ring,  is  held  by  the  latter  with  its  axis 
parallel  to  the  frame  and  pointing  to  the  horizon-glass.  The  dis- 
tance to  the  axis  from  the  frame  is  so  regulated  that  the  objective 
will  receive  rays  passing  through  the  unsilvered,  as  well  as  rays 
reflected  from  the  silvered,  part  of  the  horizon-glass.  Since  each 
portion  of  an  objective  forms  as  perfect  an  image  as  does  the  whole, 
the  difference  being  only  in  degree  of  brightness  of  the  image,  it  is 
manifest  that  by  pointing  the  telescope  at  one  object  and  placing 
another  so  that  its  reflected  rays  will  be  received  by  the  objective, 
an  image  of  each  object  may  be  seen  in  the  field  of  view,  each  per- 
fect in  detail,  but  less  bright  than  if  formed  with  the  whole  aper- 
ture of  the  objective.  The  relative  brightness  of  the  two  images 
may  be  varied  at  will  by  simply  moving  the  telescope  bodily  to  or 


PRACTICAL  ASTKONOMY. 


TO  FIND  THE  TIME  BY  ASTRONOMICAL  OBSERVATIONS.     53 

from  the  frame,  thus  presenting  more  or  less  of  the  objective  to  the 
silvered  part  of  the  horizon-glass.  For  observation,  they  should  be 
equally  bright. 

Excessive  brightness,  as  in  case  of  the  sun,  is  reduced  by  two 
sets  of  colored  shades  of  different  degrees  of  opacity,  one  set  for  the 
reflected,  and  one  for  the  direct  rays.  These  are  supposed  to  be 
of  plane  glass,  but  to  eliminate  any  errors  due  to  a  possible  pris- 
matic form,  they  admit  of  easy  reversal.  A  disk  containing  a  set 
of  colored  glasses  is  arranged  to  screw  over  the  eye  end  of  the  tele- 
scope. This  should  be  used  when  practicable,  since  any  prismatic 
form  in  these  glasses  will  aifect  both  direct  and  reflected  rays 
equally. 

Two  parallel  wires  are  placed  in  the  focus  of  the  objective,  the 
middle  point  between  which  marks  the  center  of  the  field  of  view. 
The  line  joining  this  point  and  the  optical  center  of  the  objective 
is  the  axis  of  the  telescope.  It  is  this  line  which  should  be  parallel 
with  the  frame  of  the  instrument. 

Suppose  now  with  the  index-arm  set  at  zero  (in  which  case  the 
mirrors  are  parallel),  the  telescope  is  accurately  directed  to  some 
very  distant  point.  Rays  will  pass  through  the  unsilvered  part  of 
the  horizon  glass  and  form  an  image  at  the  center  of  the  field  of 
view.  Eays  sensibly  parallel  to  these  will  fall  upon  the  index-glass, 
be  reflected  to  the  horizon-glass,  and  thence  into  parallelism  with 
the  original  direction,  since  the  angle  between  the  mirrors  is  zero. 

These  reflected  rays  being  parallel  to  the  direct  rays,  will  be 
brought  to  the  same  focus,  and  there  will  be  presented  at  the  mid- 
dle of  the  field  of  view,  apparently  one,  in  reality  two,  images  of  the 
point,  accurately  coinciding. 

Eetaining  the  direct  image  at  tho  middle  of  the  field,  let  the 
index-arm  be  moved  forward,  say  25°.  According  to  the  principle 
of  Optics  cited,  there  will  be  superimposed  on  the  first  image  that 
of  another  point,  separated  from  the  first  point  by  an  angular  dis- 
tance of  50°.  Accordingly  in  order  to  give  the  real  value  of  an 
angle,  the  sextant  graduations  are  marked  double  their  true  value. 

Also  according  to  the  same  principle  of  Optics,  it  fo/lows  that  if 
the  reading  is  50°  when  the  distance  is  50°,  the  ray  from  the  second 
point  and  all  its  reflections  must  determine  a  plane  perpendicular 
to  both  mirrors  and  hence  parallel  to  the  frame.  If  the  instrument 
and  index-arm  be  so  moved  as  to  produce  coincidence  of  images  on 


54  PRACTICAL  ASTRONOMY. 

either  side  of  the  field,  evidently  the  last  direction  of  the  ray  is  not 
parallel  to  the  frame,  the  fundamental  principle  of  the  sextant  is 
violated,  and  the  position  assumed  by  the  index-arm  to  give  this 
coincidence  gives  an  incorrect  value  of  the  angle.  The  frame  of 
the  instrument  must  therefore  always  be  held  in  the  plane  of  the 
two  points,  which  condition  is  fulfilled  when  coincidence  of  their 
images  can  be  produced  at  the  centre  of  the  field. 

Hence,  to  measure  an  angle  with  a  sextant:— Direct  the  tele- 
scope to  the  fainter  of  the  two  objects  and  bring  its  image  to  the 
middle  of  the  field.  Retaining  it  in  this  position,  rotate  the  instru- 
ment about  the  line  of  sight  and  move  the  index-arm  slowly  back 
and  forth  until  accurate  coincidence  of  the  two  images  is  produced 
at  the  middle  of  the  field.  Perfection  of  coincidence  is  produced 
by  use  of  the  tangent  screw. 

In  measuring  altitudes  (e.g.  of  the  sun)  at  sea,  it  is  sufficient  to 
bring  the  reflected  image  tangent  to  the  sea  horizon,  and  correct 
the  resulting  altitude  for  dip.  On  land  the  natural  horizon  cannot 
be  used  for  obvious  reasons.  Recourse  is  therefore  had  to  an  "  arti- 
ficial horizon  "  consisting  of  a  small  vessel  of  mercury  with  its  sur- 
face protected  against  wind,  etc.,  by  a  glass  roof.  An  observer 
placing  himself  in  the  plane  of  the  "  object "  and  the  perpendicular 
to  the  artificial  horizon,  will  by  placing  the  eye  at  the  proper  angle 
see  an  image  of  the  object  reflected  from  the  mercury.  Since  the 
angles  of  incidence  and  reflection  are  equal,  this  image  may  be  re- 
garded as  another  body  at  the  distance  of  the  object  and  at  the  same 
angular  distance  Mow  the  horizon  as  the  real  object  is  above  it. 
The  measurement  of  the  angle  between  the  two  will  therefore  give 
the  double  altitude  of  the  object.  This  measurement  is  accom- 
plished by  regarding  the  image  seen  in  the  mercury  as  the  "  fainter 
of  the  two  objects  "  mentioned  in  the  foregoing  rule,  and  then  pro- 
ceeding as  there  indicated. 

If  the  body  have  a  sensible  diameter,  as  the  sun,  the  altitude  of 
the  center  is  the  quantity  sought,  since  all  data  in  the  Ephemeris 
relating  to  the  sun  is  given  for  its  center.  Nevertheless  since  it  is 
easier  to  judge  of  the  exact  tan  gen  cy  of  the  two  images  than  of 
their  exact  coincidence,  it  is  the  altitude  of  a  limb  which  is  always 
measured.  This,  corrected  for  refraction,  semi-diameter,  and  par- 
allax, will  give  the  true  geocentric  altitude  of  the  center. 

The   sextant   being  usually  held   in  the   hand  and  therefore 


TO  FIND  THE  TIME  BY  ASTRONOMICAL  OBSERVATIONS.     55 

somewhat  unstable,  being  also  of  small  dimensions  and  graduated 
on  the  arc  only  to  10',  a  single  measurement  of  an  angle  never 
suffices  for  any  astronomical  purpose.  Altitudes  are  therefore 
always  taken  in  "sets  "  and  the  corresponding  times  noted.  There 
are  two  methods  of  taking  these  sets  according  as  the  body  is  mov- 
ing rapidly  or  slowly  in  altitude.  The  first  case  evidently  applies 
to  extra-meridian  observations,  and  the  second  to  circum-meridian 
and  circum-polar  observations.  % 

To  explain  the  first  case,  suppose  it  were  required  to  take  a  set 
of  forenoon  altitudes  of  the  sun's  upper  limb.  First  it  is  to  be 
noted  that  the  image  in  the  horizon  as  viewed  by  the  telescope,  is 
erect;  it  having  been  inverted  once  by  the  reflection  and  again  by 
the  telescope.  The  image  reflected  from  the  mirrors  is  however 
inverted,  and  its  lowest  point  corresponds  to  the  upper  limb  of  the 
sun.  Now  point  the  telescope  to  the  mercury  (having  applied  the 
proper  shades),  and  place  the  upper  limb  of  this  image  at  the  center 
of  the  field.  By  the  rotation  and  movement  of  the  index-arm  be- 
fore described  bring  the  image  from  the  mirrors  into  the  field 
above  the  other.  Since  the  sun  is  rising,  this  image  (inverted)  will 
appear  to  be  slowly  falling  in  the  field  of  view  toward  the  other. 
Set  the  vernier  at  the  nearest  outward  exact  division  of  the  limb, 
and  note  the  instant  when  the  two  images  are  just  tangent.  Set 
the  vernier  at  the  next  exact  outward  division  of  the  limb  (which 
operation  separates  the  images),  and  note  again  the  time  when  they 
come  to  tangency,  which  will  be  only  a  few  seconds  later.  So  pro- 
ceed until  the  set  is  complete.  The  altitudes  are  thus  equidistant, 
involve  no  reading  of  the  vernier,  and  while  waiting  for  contact 
the  instrument  can  be  held  steady  by  both  hands. 

To  take  altitudes  of  the  lower  limb,  allow  the  falling  image  to 
pass  over  the  other  and  note  the  instants  of  separation. 

In  the  afternoon,  the  image  here  described  as  falling,  is  rising. 

In  the  second  case,  when  the  body  is  about  to  pass  the  meridian 
or  is  near  the  pole,  it  is  moving  so  slowly  in  altitude  that  we  can- 
not set  the  index-arm  ahead  by  successive  equal  steps  and  wait  for 
the  body  to  reach  that  altitude.  Moreover  upon  passing  the 
meridian  the  motion  in  altitude  is  reversed.  In  this  case  we  must 
therefore  measure  the  altitudes  of  the  selected  limb  in  as  quick 
succession  as  possible  according  to  the  ordinary  method. 

The  same  principles  apply  in  case  of  a  star. 


56  PRACTICAL  ASTRONOMY.- 

The  glass  forming  the  roof  of  the  horizon  may  be  somewhat 
prismatic.  The  effect  of  this  may  be  eliminated  by  taking  another 
set  with  the  roof  reversed. 

ADJUSTMENTS  OF  THE  SEXTANT. 

Hitherto  it  has  been  assumed  that  both  mirrors  were  accurately 
perpendicular  to  the  frame,  that  when  they  were  parallel  to  each 
other  the  index-arm*  vernier  reads  zero,  that  the  center  of  motion 
of  the  arm  was  the  center  of  the  graduated  limb,  and  that  the 
telescope  axis  was  parallel  to  the  frame.  The  mirrors  and  telescope 
are  however  not  rigid  in  their  connections,  but  each  is  susceptible 
of  a  slight  motion  to  perfect  the  adjustment.  Well-known  optical 
principles  together  with  the  preceding  remarks  render  any  expla- 
nation of  these  adjustments  unnecessary. 

1st.  Adjustment: — To  make  the  index-glass  perpendicular  to  the 
frame. 

Set  the  index  near  the  middle  of  the  arc  ;  remove  the  telescope 
and  place  the  eye  near  the  index-glass  nearly  in  the  plane  of  the 
frame.  Observe  at  the  right-hand  edge  of  the  glass  whether  the 
arc  as  seen  directly  and  its  reflected  image  form  one  continuous 
arc,  which  can  only  be  the  case  when  the  glass  is  perpendicular. 
If  not,  tip  the  glass  slightly  by  the  proper  screws  until  the  above 
test  is  fulfilled. 

2d.  Adjustment : — To  make  the  horizon-glass  perpendicular  to  the 
frame. 

The  first  adjustment  having  been  perfected,  the  second  is  tested 
by  noting  whether  the  two  mirrors  are  parallel  for  some  one  position 
of  the  index-glass.  If  so,  the  horizon-glass  must  also  be  perpen- 
dicular to  the  frame. 

Point  the  telescope  to  a  3d  or  4th  magnitude  star,  or  to  a  distant 
terrestrial  point,  the  plane  of  the  frame  being  vertical.  Move  the 
index-arm  slowly  back  and  forth  over  the  zero.  This  will  cause 
the  reflected  image  to  pass  through  the  field  ;  if  it  passes  exactly 
over  the  direct  image  the  two  mirrors  must  be  perpendicular  to  the 
frame.  If  it  passes  to  one  side,  tip  the  horizon-glass  by  the  proper 
screws  until  the  test  is  fulfilled. 

3d.  Adjustment: — To  make  the  axis  of  the  telescope  parallel  to 
the  frame. 

Turn  the  telescope  until  the  wires  before  referred  to  are  parallel 


TO  FIND  THE  TIME  BY  ASTRONOMICAL  OBSERVATIONS.     57 

to  the  frame.  (An  adjusting  telescope  in  which  the  wires  are  well 
separated  is  to  be  preferred.)  Select  two  objects  which  are  at  a 
considerable  distance  apart,  as  the  sun  and  moon  when  distant  100°  or 
more  from  each  other.  Point  the  telescope  to  the  moon  and  bring 
the  image  of  the  sun  tangent  to  it  on  one  of  the  wires.  Move  the 
instrument  till  the  images  appear  on  the  other  wire.  If  the  tangency 
still  exists,  the  telescope  is  adjusted.  Otherwise  tip  the  ring  hold- 
ing it,  by  means  of  the  proper  screws,  till  the  test  is  fulfilled. 

4th.  Adjustment: — To  make  the  mirrors  parallel  when  the  read- 
ing of  the  arc  is  zero. 

Set  the  index  exactly  at  zero  and  point  to  the  distant  object 
described  in  the  second  adjustment.  If  the  two  images  are  exactly 
coincident,  the  adjustment  is  perfect.  Otherwise  turn  the  horizon- 
glass  around  an  axis  perpendicular  to  the  frame,  by  the  proper 
screws,  until  coincidence  is  secured.  The  mirrors  are  now  parallel. 

ERRORS  OF  THE  SEXTANT. 

It  should  be  remembered  that  to  whatever  division  of  the  arc 
the  index  may  point  ivlien  the  mirrors  are  parallel,  this  division  is 
the  temporary  zero,  and  from  it  all  angle  readings  must  be  reckoned. 
The  fourth  adjustment  will  not  remain  perfect;  it  is  therefore 
easier  to  determine  the  temporary  zero  from  time  to  time,  note  its 
distance  and  direction  from  the  zero  of  the  graduation,  and  apply 
the  correction  to  all  readings.  The  distance  in  arc  of  the  tempo- 
rary from  the  fixed  zero  is  called  the  "  Index  Error,"  positive  if  the 
temporary  zero  lie  beyond  the  graduated  arc,  negative  if  on.  To 
facilitate  its  measurement  when  positive,  the  graduations  are  car- 
ried 4  to  5  degrees  to  the  right  "of  the  zero,  constituting  what  is 
called  the  "  extra  arc." 

To  measure  the  index-error,  bring  the  mirrors  to  parallelism  by 
producing  a  perfect  coincidence  of  the  direct  and  reflected  images 
of  a  star  or  distant  point;  read  the  vernier,  giving  the  result  the 
proper  sign. 

Another  method  specially  applicable  at  sea  is  as  follows : 

Measure  the  horizontal  diameter  of  the  sun  (so  that  the  two 
limbs  may  not  be  affected  by  unequal  refraction),  first  on  the  arc 
and  then  on  the  extra  arc.  Evidently  one  reading  will  exceed,  and 
the  other  be  less  than  the  diameter,  by  the  index-error.  One  half 


58  PRACTICAL  ASTRONOMY. 

the  difference  will  then  be  the  index  error,  positive  if  the  larger 
reading  be  on  the  extra  arc. 

As  a  verification,  one  fourth  the  sum  should  be  the  sun's  semL 
diameter  as  given  for  the  date  in  the  Epherneris. 

Another  error  which  must  be  attended  to  with  equal  care  is  the 
"  Eccentricity."  This  arises  when  jjie  center  of  motion  of  the  in- 
dex-arm is  not  coincident  with  the  center  of  the  graduated  arc. 
The  effect  of  such  maladjustment  is  seen  from  Figure  15,  a  being 

the  center  of  motion,  and  b  that  of 
the  arc.  When  the  arm  is  in  the  posi- 
tion ae  in  prolongation  of  the  line 
joining  the  two  centers,  there  is  mani- 
festly no  error  in  the  reading.  When 
at  ad  perpendicular  to  that  line,  there 
is  an  error  cd.  Between  these  two 
FIG  15  positions  the  error  will  be  intermedi- 

ate in  value. 

To  determine  this  error,  measure  with  a  theodolite  the  angular 
distance  between  two  distant  points.  Then  take  the  mean  of 
several  measurements  of  the  same  angle  with  the  sextant.  The 
difference  will  be  the  effect  of  eccentricity  for  that  reading  of  the 
sextant.  This  operation  should  be  repeated  at  short  angular  inter- 
vals for  the  whole  arc,  and  the  results  tabulated. 

Other  methods  may  be  adopted  when  the  appliances  of  a  fixed 
observatory  are  at  hand. 


Nomenclature  of  the  Astronomical  Triangle. 

A     =  azimuth  angle       —  angle  at  the  zenith. 

P    =  hour  angle  =  angle  at  the  pole. 

?/>     —  parallactic  angle  =  angle  at  the  body. 

90°  —  0  =  side  from  zenith  to  pole. 

90°  —  #  =  d  =  side  from  pole  to  body  =  polar  distance. 

90°  —  a  =  z  —  side  from  zenith  to  body  —  zenith  distance. 

In  which 

0     =  latitude  of  place. 
d     =  declination  of  body. 
a      —  altitude  of  body. 


TO  FIND  THE  TIME  BY  ASTRONOMICAL  OBSERVATIONS.     59 


II.    TIME    BY    SINGLE    ALTITUDES. 

1.  To  Find  the  Error  of  a  Sidereal  Time-piece  by  a  Single  Altitude 
of  a  Star.  (See  Form  3.) — The  solution  of  this  problem  consists  in 
finding  the  value  of  the  hour  angle  Z  P  S  in  the  astronomical  tri- 
angle (see  Fig.  10),  having  given  the  three  sides  of  the  triangle, 
viz.:  Z  P,  the  complement  of  the  latitude,  P  S  the  polar  distance 
of  the  star,  and  Z  8  its  zenith  distance.  The  latitude  0  is  sup- 
posed to  be  known,  the  polar  distance  d  is  taken  from  the 


Ephemeris  for  the  date,  and  the  altitude  a,  the  complement  of  the 
zenith  distance,  is  measured  by  the  sextant  and  artificial  horizon. 
The  measured  altitude  having  been  corrected  for  errors  of  the  sex- 
tant and  refraction,  the  above  da.ta  substituted  in  the  formula 


sin  i  P  = 


cos  m  sin  (m  —  a] 
cos  0  sin  d       ' 


(62) 


will  give  the  value  of  P,  the  star's  hour  angle,  which  divided  by  15 
will  give  the  hour  angle  in  time.  (The  negative  sign  is  to  be  used 
if  the  star  be  east  of  the  meridian.) 

This  plus  the  star's  R.  A.  for  the  date  will  give  the  sidereal 
time,  which  by  comparison  with  the  chronometer  time  noted  at  the 
Instant  of  taking  the  altitude,  will  give  the  chronometer  error. 

As  heretofore  stated,  reliance  is  not  to  be  placed  upon  a  single 
measurement  by  so  defective  an  instrument  as  the  sextant.  A  set 
of  observations,  from  5  to  10,  is  therefore  made  by  recording  the 
times  corresponding  to  successive  changes  of  10'  in  the  star's 
double  altitude.  These  altitudes  will  thus  be  equidistant  and  in- 
volve no  measurement  of  seconds  of  arc. 


60  PRACTICAL  ASTRONOMY. 

In  the  computations  it  is  usual  to  assume  that  the  mean  of  the 
times  corresponds  to  the  mean  of  the  altitudes,  as  shown  on  Form 
3,  which  implies  that  the  star's  motion  in  altitude  is  uniform.  This 
in  general  is  not  true.  We  must  therefore,  to  be  as  accurate  as 
possible,  either  apply  a  correction  to  the  mean  of  the  times  to 
obtain  the  time  when  the  star  was  at  the  mean  of  the  altitudes,  or 
a  correction  to  the  mean  of  the  altitudes  to  give  the  altitude  at  the 
mean  of  the  times.  Whether  corrected  or  not,  the  means  are  used 
as  a  single  observation.  Also,  since  the  refraction  raries  ununi- 
formly  with  the  altitude,  the  refraction  correspondii \g  to  the  mean 
of  the  altitudes  requires,  in  strictness,  a  slight  correction;  although 
of  much  less  importance  than  the  first.  These  corrections  may  as 
a  rule  be  omitted.  Their  deduction  is  given  in  the  following  para- 
graph. 


*%»  To  determine  the  correction  to  be  applied  to  the  mean  of  the 
altitudes  or  the  mean  of  the  times,  the  following  deduction  is  ap- 
pended essentially  as  given  by  Chauvenet. 

To  find  the  change  in  altitude  of  a  star  in  a  given  interval  of 
time,  having  regard  to  second  differences,  let 


Then  a  +  A  a  =f(P  +  A  P). 

Expanding  by  Taylor's  Theorem, 


From  the  astronomical  triangle, 

sin  a  —  cos  d  sin  0  -|-  sin  d  cos  0  cos  P. 
cos  a  d  a  =  —  sin  d  cos  0  sin  P  d  P. 

da  sin  d  cos  0  sin  P  , 

dP  =  ---  55TF-        =-cos0sm^ 
A  being  the  azimuth. 

-cos0cosJ-.  (64) 


TO  FIND  THE  TIME  BY  ASTRONOMICAL  OBSERVATIONS.     61 
Also  from  the  astronomical  triangle  in  a  similar  manner, 
d  A  cos  ft  sin  A 


dP  sin  P      ' 


(65) 


being  the  parallactic  angle. 
Whence 


,  cos  0 sin  A  cos  ^4  cos  ft  (A P)a   , 

A  a  =  —  cos  0  sin  J[  AP  -| —  — : — 73 — ~ .  (66) 

sin  _t  /v 

Expressing  A  a  and  A  P  in  seconds  of  arc  and  time  respec- 
tively, we  have,  after  reduction, 

A  a  —  —  cos  0  sin  A  (15  A  P) 

cos  0  sin  A  cos  ^  cos  ft   (15AP)a     .       „          , 
sinTP  2 

which  gives  the  variation  in  altitude  due  to  a  lapse  of  A  P  seconds 
of  time. 

The  last  term  may  be  written 


2  sin  1 

Values  of  m  are  given  in  tables  under  the  head  of  Eeduction  to  the 
Meridian. 

Placing  also,  for  brevity, 

.      7       cos  A  cos  0 
g  =  cos  0  sm  -4,     *  =  —  ^-^  —  , 

we  have, 

A  a  =  — 

a  more  convenient  expression  of  the  same  relation. 

Now_  let  H,  H',  H",  etc.,  denote  the  altitudes  (corrected  for 
sextant  errors),  T,  T',  T",  etc.,  the  corresponding  times,  aa  the 
mean  of  the  altitudes,  t0  the  mean  of  the  times,  and  a0f  the  altitude 
corresponding  to  t0  ,  since  this  cannot  be  a0.  It  is  now  required  to 
determine  the  relation  between  a/  and  aQ  in  order  that  the  whole 


62  PRACTICAL  ASTRONOMY. 

set  of   observations  may  be  resolved  into  one  —  a  single  altitude 
taken  at  the  mean  of  the  times. 

The  change  H  —  a0r  required  the  time  T  —  t0. 
The  change  H'  —  a0'  required  the  time  T'  —  t0  , 

etc.- 

Therefore   from   the   relation  A  a  =  —  15  ^  A  P  -f  g  k  m  we 
have,  denoting  the  different  m's  by  w,  ,  w2,  etc., 


0iy 
H'  -a=-l$T'  -t 


etc.  etc. 

If  there  were  n  observations,  the  mean  gives. 

,  m,  4-  m0  +  m.  4-  etc.  ,__, 

a.  -a.  '  =  gk-  l-^~  -=gkm,.  (69) 

Or 

«„'  =  ao~9k  m0.  (70) 

The  last  term  is  therefore  the  desired  correction  to  the  mean  of 
the  altitudes  in  order  that  it  may  correspond  to  the  mean  of  the 
times. 

It  will  however  be  more  convenient  to  find  such  a  correction  as 
applied  to  the  mean  of  the  times  will  cause  it  to  correspond  to  the 
mean  of  the  altitudes. 

Let  tQ'  denote  the  time  corresponding  to  the  mean  of  the  alti- 
tudes. 

The  change  a0  —  a0f  required  the  time  t0'  —  t0.  Hence  from 
the  preceding,  we  have,  since  t0'  —  t0  is  very  small, 


7       cos  A  cos  ^  .    ,  .... 

Expressing  k  =  --  :  —  ^—L  in  known  quantities, 


sin 


gin  feog^Binrf  ring  _  ^  p 

cos  a  v     ' 


TO  FIND  THE  TIME  BY  ASTRONOMICAL  OBSERVATIONS.     68 

t>  =  tt  +  TVrcot  P  -  **r«»*'*»<V  3  «^A^).  (T» 

1  5  1_  cos  «0  cot  «0     J  n  sin  1" 

The  refraction,  r,  belonging  to  the  mean  of  the  altitudes  is  cor- 
rected, if  desired,  by  the  quantity 

sin  r  1   „  2  sin2  \  (an  —  H) 
sin'J  a0  n  sin  1" 

77  denoting  the  different  altitudes. 


It  is  important  to  ascertain  what  stars  are  suited  to  the  solution 
of  this  problem. 

Differentiating  the  equation  derived  from  the  astronomical  tri- 
angle (regarding  a  and  P  as  variable), 

sin  a  =  sin  <p  sin  d  -f  cos  0  cos  d  cos  P,  (73  ) 

and  reducing  by 

cos  a  _  sin  P 

cos  6  ~  sin  A9 
we  have 

dP  =  ----  -U  —  -  d  a.  (74) 

cos  0  sin  ^4 

From  this  it  is  seen  that  any  error  (da)  in  the  measured  alti- 
tude will  have  the  least  effect  on  the  computed  hour  angle  when 
0  =  0°,  and  A  —  90°.  That  is,  the  method  is  less  exposed  to  error 
in  low  latitudes;  but  whatever  the  latitude,  the  star  should  be  near 
the  prime  vertical.  The  worst  position  of  the  star  is  when  on  the 
meridian. 

Differentiating  the  same  equation  regarding  0  and  P  as  variable, 
reducing  by 

cos  a  cos  A  =  sin  d  cos  0  —  cos  d  sin  0  cos  P.  (75) 

and  the  same  equation  as  before,  we  have 


cos  0  tan  A 


64  PRACTICAL  ASTRONOMY. 

From  this  it  is  seen  that  any  uncertainty  as  to  the  exact  latitude 
will  also  have  least  effect  when  the  star  is  near  the  prime  vertical 
and  the  observer  near  the  equator. 

Differentiating  with  reference  to  6  and  P,  we  have 

dd,  (77) 


cos  6  tan 

and  it  thus  appears  that  an  erroneous  value  of  d  will  also  produce 
the  least  effect  when  the  star  is  on  the  prime  vertical,  since  from 
the  equation 

.          COS  0    .        A 

sin  ib  =  -  2:  sm  A 
cos  d 

sin  if}  and  therefore  tan  if?  will  be  a  maximum  when  sin  A  is  also 
a  maximum. 

From  the  three  foregoing  differential  equations  it  is  also  seen 
that  the  effect  of  constant  errors  either  in  the  measured  altitude, 
the  assumed  latitude,  or  assumed  declination,  may  be  eliminated 
by  combining  the  results  from  two  stars,  one  east  and  one  west  of 
the  meridian,  and  in  as  nearly  corresponding  positions  as  possible; 
since  then  the  corresponding  values  of  sin  A,  tan  A9  and  tan  fy  will 
be  numerically  nearly  equal  and  of  opposite  signs. 

Hence  the  following  general  rule  should  be  observed  :  In  order 
to  reduce  to  a  minimum  the  effect  of  errors  either  in  the  observations 
or  the  assumed  data,  select  a  star  which  wilt  cross  the  prime  vertical 
at  some  distance  from  the  zenith  (S  <  0),  and  make  the  observations 
near  that  circle.  As  the  latitude  increases,  greater  accuracy  in  the 
observations  and  data  is  required  in  ordsr  to  give  a  constant  degree* 
of  precision  in  the  results.  Stars  very  near  the  horizon  should  be 
avoided  on  account  of  excessive  and  irregular  refraction.  TJie 
adopted  value  of  the  clock  error  should  be  the  mean  of  the  results 
from  an  east  and  a  west  star. 

In  the  computation,  if  great  accuracy  be  not  essential,  mean  re- 
fractions may  be  employed  ;  their  values  are  given  in  tables. 

2.  To  Find  the  Error  of  a  Mean  Solar  Time-piece  by  a  Single  Alti- 
tude of  the  Sun's  Limb.  (See  Form  4.)—  This  problem  does  not 
differ  in  principle  from  the  preceding.  The  observations  are  made 
on  the  sun's  limb,  and  therefore  in  addition  to  refraction  the  cor- 


TO  FIND  THE  TIME  BY  ASTRONOMICAL  OBSERVATIONS.     65 

rection  for  semi-diameter  at  the  time  of  observation  must  be  ap- 
plied. Also,  since  the  sun  has  an  appreciable  parallax,  and  since 
also  the  Ephemeris  data  supposes  the  observer  to  be  at  the  earth's 
center,  the  altitude  must  be  further  corrected  for  "  parallax  in  alti- 
tude." Parallax  in  altitude  =  Equatorial  Horizontal  Parallax  X  p 
X  cos  altitude,  p  being  the  ratio  of  the  earth's  radius  at  the  equator 
to  that  at  the  place  of  observation.  At  West  Point  log  p  =  9.999368 
—  10.  The  Equatorial  Parallax  is  given  in  the  Ephemeris,  page 
278. 

The  sun's  declination  (or  polar  distance)  which  is  given  in  the 
Ephemeris  for  certain  instants  of  Greenwich  time,  varies  quite  rap- 
idly; and  in  order  to  determine  this  element  at  the  instant  of  ob- 
servation we  must  know  our  longitude  and  the  error  of  the  chro- 
nometer, to  obtain  which  is  the  object  of  the  problem.  In  practice, 
however,  the  error  will  usually  be  known  with  sufficient  accuracy 
to  find  approximately  the  time  elapsed  since  Greenwich  mean  noon. 
With  this  assumed  difference  we  find  by  interpolation  in  page  II, 
Monthly  Calendar,  the  declination  for  the  instant.  The  same  re- 
marks apply  to  the  determination  of  the  semi-diameter  referred  to 
above,  and  the  Equation  of  Time  below. 

With  the  data  thus  found,  compute  P  (in  time)  as  in  the  pre- 
ceding problem. 

Then  if  it  be  a  morning  observation, 

Apparent  time  —  12h  —  P. 
If  an  afternoon  observation, 

Apparent  time  =  P. 

Apparent  time  ±  Equation  of  Time  =  Mean  Time.  This  com- 
pared with  the  mean  of  the  recorded  times  gives  the  chronometer 
error,  and  if  this  is  found  to  diifer  very  materially  from  the  assumed 
error,  the  declination  and  possibly  also  the  Semi-diameter  and 
Equation  of  Time,  must  be  redetermined,  and  the  computation  re- 
peated. The  sun  should  be  observed  as  near  the  prime  vertical  as 
is  consistent  with  avoiding  irregular  refraction. 

In  all  cases  where  time  is  to  be  determined  by  altitudes  of  the 
sun,  it  is  better  to  make  a  set  of  observations  on  each  limb,  and  re- 


66  PRACTICAL  ASTRONOMY. 

duce  each  set  separately.  If  a  difference  of  personal  error  in  esti- 
mating contact  of  images  as  compared  with  their  separation  exists, 
it  will  thus  be  discovered  and  in  a  great  measure  eliminated. 

III.   TIME.  BY   EQUAL   ALTITUDES. 

1.  To  Find  the  Error  of  a  Sidereal  Time-piece  by  Equal  Altitudes 
of  a  Star.  (See  Form  5.) — If  the  times  when  a  star  reaches  equal 
altitudes  on  opposite  sides  of  the  meridian  be  noted,  the  "  middle 
chronometer  time"  will  be  the  time  of  transit,  provided  the  chro- 
nometer has  run  uniformly.  Hence  we  would  have 

T  -J-  T 
E=a-     '  T    "•  (78) 

6 

But  if  the  refraction  is  different  at  the  times  of  the  two  observa- 
tions, the  true  altitudes  will  be  unequal  when  the  observed  are 
equal;  which  latter  will  consequently  not  correspond  to  equal  hour 
angles.  Manifestly  therefore  one  of  the  chronometer  times  (e.g., 
the  last),  requires  a  correction  equal  to  the  hour  angle  correspond- 
ing to  the  change  in  true  altitude, — this  change  being  the  difference 
between  the  E.  and  W.  refractions, — and  the  middle  chronometer 
time  will  require  one  half  this  correction. 

Hence  we  have  in  full  (see  note  at  end  of  next  problem), 


f1  —   a  _  e     '          "    _]_  _         V'g  '  Wl    ^""   ">  /»~g\ 

L       2         r  215  cos  0  cos  tf  sin  tj 

E  being  the  chronometer  error  at  time  of  meridian  passage,  a  the 
star's  apparent  R.  A.,  Te  and  Tw  the  chronometer  times  of  observa- 
tion, re  and  rw  the  east  and  west  refractions,  and  t  one  half  the 
elapsed  time  between  the  observations.  The  above  equation  evi- 
dently applies  even  when  the  times  have  been  noted  by  a  mean 
solar  chronometer,  provided  a  be  replaced  by  the  computed  mean 
time  of  meridian  passage. 

Use  an  Ephemeris  star  and  make  the  first  set  of  observations  as 
prescribed  under  "  Time  by  Single  Altitudes."  Then  with  the  same 
sextant  use  the  same  altitudes  in  the  second  set,  of  course  in  the 
reverse  order. 


TO  FIND  THE  TIME  BY  ASTRONOMICAL  OBSERVATIONS.     67 

From  the  preceding  Equation  it  is  seen  that  the  actual  altitudes 
are  not  required.  Therefore  unless  the  correction  for  refraction 
is  to  be  applied,  no  record  need  be  made  of  the  sextant  readings  or 
errors.  Also,  under  the  same  condition,  the  method  is  independent 
of  errors  in  the  assumed  latitude  or  the  star's  declination. 

As  before,  the  observations  should  be  made  as  near  the  prime 
vertical  as  is  consistent  with  avoiding  irregular  refraction.  By 
selecting  a  star  whose  declination  is  but  a  little  less  than  0,  it  will 
be  on  the  prime  vertical  near  the  zenith,  and  we  can  probably  avoid 
the  correction  for  refraction  since  the  elapsed  time  will  be  small. 
The  sextant  and  chronometer  also  will  be  but  little  liable  to  changes. 

If  the  eastern  observations  have  been  prevented  by  clouds  or 
other  cause,  we  may  still  take  the  .western  observations,  and  the 
eastern  at  the  next  prime  vertical  transit  of  the  star;  thus  giving 
the  chronometer  error  at  time  of  star's  lower  meridian  passage. 

2.  To  Find  the  Error  of  a  Mean  Solar  Time-piece  by  Equal  Alti- 
tudes of  the  Sun's  Limb.  (See  Form  6.)  —  The  general  principles 
involved  and  the  methods  of  observation  are  the  same  as  in  the  pre- 
ceding problem.  But  since  the  sun  changes  in  declination  between 
the  times  of  the  E.  and  ^V.  observations,  equal  altitudes  do  not  cor- 
respond to  equal  hour  angles.  For  example,  when  the  sun  is  mov- 
ing north,  the  morning  will  be  less  than  the  afternoon  hour  angle 
at  the  same  altitude.  Manifestly  therefore  the  afternoon  hour  angle 
requires  to  be  diminished  by  the  change  due  to  the  change  of  decli- 
nation, and  the  middle  chronometer  time  by  half  this  amount,  which 
is  accomplished  in  practice  by  adding  the  correction  with  its  sign 
changed.  This  correction  is  called  the  "  Equation  of  Equal  Alti- 
tudes." 

The  middle  chronometer  time  thus  corrected  gives  the  chro- 
nometer time  of  apparent  noon.  12h  ±  the  Equation  of  time  at 
Apparent  Noon  gives  the  mean  time  of  apparent  noon,  and  the  dif- 
ference is  the  chronometer  error  on  mean  time  at  apparent  noon. 

Hence  in  full 


2  15  cos  0  cos  #  sm  t 
(A  K  tan  0  +  B  K  tan  tf)    .    (80) 


68  PRACTICAL  ASTRONOMY. 

The  last  term  in  the  bracket  is  the  Equation  of  Equal  Altitudes. 
For  its  deduction,  see  note  at  end  of  problem. 
.     A  and  B  are  taken  from  tables.      K  is  the  sun's  hourly  increase, 
in  declination  at  apparent  noon,  taken  from  the  Ephemeris  by  inter- 
polation ;  d  is  the  sun's  declination  at  same  time. 

If  a  sidereal  chronometer  had  been  used,  the  above  equation 
would  evidently  still  apply,  substituting  for  12'1  ±  e  the  sun's  R.  A. 
at  apparent  noon,  and  omitting  6'1  in  the  parenthesis. 

For  the  application  of  this  method  to  midnight,  and  effect  of 
errors  in  data,  see  Note. 


*J*  Correction  for  Refraction. — To  deduce  the  correction  for  re- 
fraction employed  in  the  two  preceding  problems,  resume  the  dif- 
ferential equation  of  the  last  note, 


„    . — -  da  (numerically), 
cos  0  cos  tf  smP  J' 


which  gives  the  change  in  hour  angle  (in  arc)  for  a  change  in  alti- 
tude of  da. 

If  the  west  refraction  be  less  than  the  east,  the  sun  will,  in  fall- 
ing, reach  the  altitude  a  too  soon,  and  the  west  hour  angle  must  be 
increased.  Hence  in  this  case  the  correction  must  be  positive  and 
additive,  and  in  any  case  the  correction  with  its  proper  sign  in  time 
will  be  obtained  from  the  expression 

(re  —  rM,)  cos  a 
15  cos  0  cos  d  sin  t ' 

since  re  —  rw  is  the  change  in  altitude  da,  and  t,  or  one  half  the 
elapsed  time,  is  practically  P. 

For  the  middle  chronometer  time,  we  therefore  have 

Cor.  for  Ref.  =  1     ('•  ~ 'J  «»* 

*  15  cos  0  cos  6  sm  t 

The  equation  reduced  as  in  the  preceding  note,  gives 

d  P  =       Te  ~  TW—  (82) 

30  cos  <p  sin  A 


TO  FIND  THE  TIME  BY  ASTRONOMICAL  OBSERVATIONS.     60 

Since  re  —  rw  may  denote  an  error  in  altitude  from  any  cause  what- 
ever, it  follows  that  the  observations  should  be  made  near  the  prime 
vertical. 

Equation  of  Equal  Altitudes. — In  order  to  deduce  the  Equation 
of  Equal  Altitudes,  resume  the  equation 

sin  a  =  sin  0  sin  #  -f-  cos  0  cos  6  cos  P. 
Differentiate,  regarding  $  and  P  as  variable,  and  solving,  we  have 

,  „      sin  0  cos  8  —  cos  0  cos  P  sin  d  _ 

a  r  =  -  — — 5T- . — =7—      —  a  o, 

cos  0  cos  o  sin  P 


which  gives  the  change  in  hour  angle  due  to  a  change  d  d  in  decli- 
nation. 

Now  if  t  denote  half  the  elapsed  time  in  hours,  and  A' the  hourly 
increase  in  the  sun's  declination  at  the  middle  instant  (assumed  to 
be  apparent  noon),  we  will  have 

d  d  ^  2  t  K. 

Again  assuming  P  to  be  the  mean  hour  angle  =:  t,  and  d  to  be  the 
declination  at  the  middle  instant  (assumed  to  be  apparent  noon),  we 
shall  have  for  the  change  in  hour  angle  in  time  due  to  the  in- 
crease in  declination 

7  ,-,        ,    /tan  0      tan 
d  P  =  TT 


Since  A"  denotes  an  increase  in  declination,  the  afternoon  hour  angle 
will  be  too  large  by  the  above  quantity,  and  the  middle  chronometer 
time  too  large  by  half  the  same  quantity.  Hence  in  any  case,  the 
quantity  to  be  added  to  the  middle  chronometer  time  to  reduce  it 
to  chronometer  time  of  apparent  noon  is 

K  t  tan  0      K  t  tan  d 
15  sin  t  15  tan  t 


70  PRACTICAL  ASTRONOMY. 

Making  A  =  —  -  —  —  -,     B  =  -      —  .,    we  have 
15  sin  t  15  tan  t 

Eq.  of  Equal  Altitudes  =  A  A"  tan  0  -f  B  JTtan  tf.       (85) 

As  in  the  preceding  case,  observations  may  be  made  in  the 
afternoon  and  the  following  morning  to  obtain  the  chronometer 
error  at  midnight.  Such  a  set  may  be  regarded  as  A.  M.  and  P.  M. 
observations  respectively  made  by  a  person  at  the  other  extremity 
of  the  earth's  diameter,  and  therefore  in  latitude  —  <p. 

Hence  for  midnight  the  Eq.  would  be 

Kt  tan  0      Jftian  3 
15  sin  t          15  tan  t  ' 

Since  t  is  always  less  than  12h,  its  sine  is  always  positive.  Also 
tan  t  will  be  positive  when  t  is  less  than  Gh,  and  negative  when 
more.  From  which  it  is  seen  that  we  may  use  a  single  equation  for 
both  noon  and  midnight,  viz.  : 


by  noting  the  following  rule  as  to  signs. 

For  noon,  A  is  always  negative,  for  midnight  positive.  For 
noon  or  midnight  B  is  positive  when  the  elapsed  time  is  less  than 
12h  and  negative  when  more. 

The  effect  of  errors  in  0  and  d  is  readily  seen  by  a  differentia- 
tion of  the  Equation. 


Time  of  Sunrise  or  Sunset. — This  problem  is  precisely  similar  to 
that  of  single  altitudes,  except  that  the  altitude  of  the  sun  is  known 
and  therefore  no  observation  is  required.  The  zenith  distance  of 
the  sun's  center  at  the  instant  when  its  upper  limb  is  on  the  hor- 
izon is  assumed  to  be  90°  50',  which  is  made  up  of  90°,  plus  16' 
(the  mean  semi-diameter  of  the  sun)', plus  34'  (the  mean  refraction 
at  the  horizon).  The  resulting  hour  angle  replaces  P  in  Form  4. 

Duration  of  Twilight.— The  zenith  distance  in  this  case  is  108°, 
as  twilight  is  assumed  to  begin  in  the  morning  or  end  in  the  even- 


DURATION  OF  TWILIGHT.  71 

ing  when  the  sun's   center  is  18°  below  the  horizon.     (See  Art. 
130,  Young.) 

From  the  solution  of  the  Z  P  S  triangle  it  can  readily  be  shown 
that  the  time  required  for  the  sun  to  pass  from  the  horizon  to  a 
zenith  distance  z  is 


. 

15  2  cos  0 


(87) 


in  which  *p  and  ip'  (called  the  sun's  parallactic  angles)  are  the  an- 
gles included  between  the  decimation  and  vertical  circles  through 
the  sun's  center  for  any  zenith  distance  z,  and  for  the  horizon  re- 
spectively, and  0  is  the  observer's  latitude.  Making  z  equal  to  108° 
this  becomes 


from  which  the  duration  of  twilight  for  any  latitude  and  any  sea- 
son of  the  year  can  be  found;  the  values  of  (p  and  ip'  are  given  by 

sin  0  —  sin  d  cos  z 

cos  ib  =  -          — ^r— •—       —,  (89) 

cos  o  sin  z 

and 

, ,      sin  0 


COS  6 


(90) 

^       ' 


When  ip  is  equal  to  ip'  then  Hs  a  minimum,  and  we  have,  after 
replacing  1  -  cos  18°  by  2  sin2  9C 


.2  no 
9 


t  =  •&  sin-1  (sin  9°  sec  0),  (91) 

from  which  the  duration  of  the  shortest  twilight  is  found.     Under 
the' same  condition  we  have  from  Eqs.  (89)  and  (90), 

sin  d  —  —  tan  9°  sin  0;  (92) 

from  which  the  sun's  declination  at  the  time  of  shortest  twilight  at 
any  latitude  can  be  found. 


72  PRACTICAL  ASTRONOMY. 


LATITUDE. 

Tho  latitude  of  a  place  on  the  earth's  surface  is  the  declination 
of  its* zenith.  The  apparent  zenith  is  the  point  in  which  the  plumb- 
line,  if  produced,  at  the  point  of  observation  would  pierce  the  celes- 
tial sphere.  The  central  zenith  is  the  point  in  which  the  radius  of 
the  earth,  if  produced,  would  pierce  the  celestial  sphere.  The  lati- 
tude measured  from  the  central  zenith  is  called  the  geocentric  lati- 
tude, and  that  from  the  apparent  zenith  is  called  the  astronomical 
latitude  or  simply  the  latitude.  The  difference  between  the  lati- 
tude and  the  geocentric  latitude  is  called  the  reduction  of  latitude. 

The  direction  of  the  plumb-line  is  affected  by  the  local  attrac- 
tion of  mountain  masses  on  the  plumb-bob,  or  on  account  of  the 
unequal  variations  of  density  of  the  crust  of  the  earth,  at  or  near 
the  locality  of  the  station.  The  Astronomical  latitude  is  deter- 
mined from  the  actual  direction  of  the  plumb-line,  and  therefore 
includes  all  abnormal  deviations.  The  Geographical  or  Geodetic 
latitude  is  that  which  would  result  from  considering  the  earth  a 
perfect  spheroid  of  revolution,  without  the  abnormal  deviations 
above  referred  to. 

Form  and  Dimensions  of  the  Earth.— Before  proceeding  tc  the 
latitude  problems  it  is  important  to  derive  some  necessary  formulas 
from  the  form  and  dimensions  of  the  earth.  For  this  purpose,  let 
us  assume  iiiat  the  earth  is  an  oblate  spheroid  about  the  polar  axis. 


FIG.  17. 

Let  E  P'  0  be  a  meridian  section  of  the  earth  through  the  observ- 
er's place  0\  C P'  the  earth's  axis;  EQ  the  earth's  equator  a,ad 


LATITUDE.  73 

H  H'  the  observer's  horizon.  Let  P  be  the  pole  of  the  heavens;  Z 
the  apparent  and  Z'  the  central  zenith ;  0  the  latitude  and  0'  the 
geocentric  latitude.  The  equation  of  the  observer's  meridian  re- 
ferred to  its  center  and  axes  is 

2      2       i       12^.2  27L2  1 1\€\\ 

ay  -\-  o  x  =  a  o  ,  (9o) 

in  which  a  and  b  are  the  equatorial  and  polar  radius  of  the  earth. 
Thejcoordinates  of  0  being  xf  and  y' ,  we  have  the  following  ana- 
lytical conditions. 

For  the  tangent  at  0,  coincident  with  the  horizon,  from 

and  the  normal  at  0,  through  the  apparent  zenith  Z,  from 

ay  (x  -  x')  -  Vx'  (y  -  y')  =  0.  (95) 

From  Eq.  (94),  we  have 

tan  0  A  C  =  tan  (90°  -  0)  =  ^;  (8(6) 

whence 

Substituting  in 
and  eliminating  b  by 


we  have 

.  a  cos  0 


Vl  -  e*  sin'  0' 
_  a  (1  —  ea)  sin  0 

"  V  1  —  easina0* 


(100) 


Let  5  be  the  length  of  any  portion  of  the  meridian;  then  for  the 
elementary  arc,  its  projection  on  the  major  axis  x,  is 


ds  cos  OA  C=ds$in  </>==  -dx'9  (101) 


74  PRACTICAL  ASTRONOMY. 

since  xf  is  a  decreasing  function  of  the  latitude.     Differentiating 
the  first  of  Eqs.  (100),  we  have 


(1  -  e2  sin2  0)f 
Equating  (101)  and  (102),  we  have 


(102) 


(103) 


(1  -  ea  sin2  0)1' 
and  for  any  other  latitude  0,  , 

*,,  =  «     (I-'*)**          '  (104) 

(l-«'sinf0,)f 

Let  d  0  =  1°,  then  dividing  (103)  by  (104),  we  have 

ds   =  (l-g«8in'0,)»  =  l-ja'Bin'0, 
d  st      (1  -  e«  sin2  0)1      1  -  I  «  «m   0 

which,  after  solving  with  reference  to  e2,  reduces  to 

,  _  2  _  ds  —  ds,  _  nofi. 

~3  ^5sm20-f^/sin20/ 

from  which  the  value  of  the  eccentricity  of  the  meridian  can  be 
found  when  the  measured  lengths  ds  and  ds,  of  any  two  portions 
of  the  meridian  line,  eaqh  1°  in  latitude,  and  the  latitudes  0  and 
0y  of  their  middle  points  are  known  ;  for  the  earth,  this  has  been 
found  to  be  about  0.0816967. 

To  find  the  equatorial  and  polar  radii,  we  have  from  Eq.  (103) 
after  making  d<p  =  1°, 

0  =  -(l-y*in'>)l,  (107) 


and  from  the  property  of  the  ellipse, 

5  =  a  VT^e*.  (108) 


LATITUDE.  75 

To  find  the  radius  of  curvature  R  at  any  point  of  the  meridian. 
After  substituting  the  values  of  dx9  dy,  and  d'y,  taken  from  Eqs. 
(100),  in  the  general  formula  for  radius  of  curvature, 


we  have 


R  =  a  -  i-ZL  ?!  —  ,;  (110) 

' 


and  hence  the  length  of  one  degree  of  latitude  at  any  latitude  0  is, 
„      2  TT  R2  TT  a         1  —  e* 


360        "360       _ 


fi  1  1  \ 


To  find  the  length  of  a  degree  on  a  section  perpendicular  to  the 
meridian  at  any  latitude  0  we  proceed  as  follows:  The  radius  p  of 
the  earth  at  the  observer's  place,  is  the  minor  axis,  and  the  equa- 
torial radius  a  is  the  major  axis  of  the  elliptical  section,  cut  out  of 
the  earth  by  a  plane  perpendicular  to  the  meridian  plane,  passed 
through  the  center  and  the  observer's  place. 

Squaring  and  adding  Eqs.  (100)  and  extracting  the  square  root, 
we  have  the  radius  of  the  earth  at  the  observer's  place;  or 


p  =  a         -  =  a  ,. 

1  —  e  sin  0  1  —  e  sm"  0 

* 

The  square  of  the  eccentricity  of  the  section  is 

fl«  _  p«     e*  (1  -  g«)  sin2  0, 
aa  1  -  e2  sin2  0 

which  being  substituted  for  e2  in  Eq.  (Ill)  after  making  0  =  90°, 
gives 


2  TT     ./        1  -  e2  sin2  0 
=  SCO  a  V  1^7-2  -.'!• 


76  PRACTICAL  ASTRONOMY. 

To  find  the  length  of  a  degree  of  longitude  at  any  latitude  0, 
we  know,  Eqs.  (100),  that  the  radius  of  the  parallel  is  x'  \  therefore 
we  have 


2  it   ,      2  TT  cos  0 

a  =  ^x=mavr^e^>' 


The  value  of  the  radius  of  the  earth,  at  any  latitude  0,  is  de- 
rived from  Eq.  (112)  or, 


_         1  -  2  e2  sin2  0  +  e4  sin2  0 
P  :  1  -  e2  sin2  0 

which,  for  logarithmic  reduction,  when  a  is  made  unity  may  be 
placed  under  the  form 

log  p  =  9.9992747  -f  0.0007271  cos  2  0  -  0.0000018  cos  4  0.  (115) 
From  the  figure  and  Eqs.  (100),  we  have 

,  , ,  a  cos  0 

x'  =  p  cos  0'  =  a    .       -,  (116) 

V 1  —  ej  sin2  0*  v      y 


.  .      ,,      a  (1  —  ea)  sin  0 

=  p  sm  0'  =  -^==L==. 
y  1  —  e2  sm' 


0 

Multiplying  these  equations  by  cos  0  and  sin  0  respectively,  adding 
and  reducing  we  have 


cos  (0  -  0')  =  -  i/l  -  e2  sin8  0,  (118) 


and  from  (116), 

a        cos 


Whence  by  combination  we  have 


cos  0.'  cos  (0  —  0')  =  -a-  cos  0;  (120) 


LATITUDE.  77 

and  solving  with  reference  to  p  we  have 


s?/  cos    -0r          : 

which  is  capable  of  logarithmic  computation. 

To  find  the  reduction  of  latitude  0  —  0'.     Since  0  is  the  angle 
made  by  the  normal  with  the  axis  of  x  we  have 

(Jy 
tan0=-^,  (122) 

and  irona  the  figure  we  have  » 

tan  0'  =  £  •  (123) 

iC 

Differentiating  the  equation  of  the  meridian  section  we  have 


Whence 

tan  0'  =  -5  tan  0  =  (1  -  ea)  tan  0.  (125) 

Developing  into  a  series,  we  have 

0  -  0'  =  ^p  sin  2  0  -  (^^^sin  40  +  etc.     (126) 

But  since  e  —  0.0816967  this  reduces  to 

0  —  0'  =  690".65  sin  2  0  —  1".16  sin  4  0  very  nearly.        (127) 

Latitude  Problems.  —  The  general  problem  of  latitude  consists 
in  finding  the  side  Z  P  in  the  ZP  S  triangle,  any  other  three  parts 
being  given. 

Differentiating  (73'),  regarding  first  a  and  0  and  next  P  and  0 
as  variable,  and  reducing  by  (75)  we  obtain 

&;  0  =  sec  A  d  a, 
and 

d  0  =  tan  A  cos  0  d  P, 


78  PRACTICAL  ASTRONOMY. 

Whence  observations  for  latitude  should  as  a  rule  be  made  upon  a 
body  at  or  near  the  time  of  its  culmination. 

The  following  are  the  methods  usually  employed. 

1.  By  Circumpolars. — This  depends  on  the  fact  that  the  altitude 
of  the  pole  is  equal  to  the  astronomical  latitude  of  the  place.     Let 
a  and  a'  be  the  altitudes  of  a  circumpolar  star  at  upper  and  lower 
culmination  respectively,  corrected  for  refraction  and  instrumental 
errors;  d  and  d'  the  corresponding  polar  distances,  and  0  the  lati- 
tude; then  we  have 

(j)  =  a-d,    <t>  =  a'+d',    <f>  =  J  (a  +  a')  -f|  (dr  -  d). 

The  change  from  d  to  d'  is  ordinarily  so  small  in  the  interval 
(12  hours)  between  the  observations  as  to  be  negligible;  it  is  due 
solely  to  precession  and  nutation.  This  method  is  free  from  dec- 
lination errors,  but  subject  to  changes  and  errors  in  the  refraction. 
It  is  therefore  an  independent  method,  and  is  the  one  used  in  fixed 
observatories  where  the  observations  can  be  made  with  great  accu- 
racy even  during  daylight  by  the  transit  circle.  With  the  sextant 
the  method  is  applicable  only  in  high  latitudes  during  the  winter 
so  that  both  culminations  occur  during  the  night  time.  A  star 
with  a  small  polar  distance  is  to  be  preferred,  to  avoid  irregular  re- 
fraction at  the  lower  culmination. 

The  sextant,  however,  is  not  well  adapted  to  this  method,  since 
the  least  count  of  its  vernier  is  usually  10",  and  at  culmination 
only  a  single  altitude  can  be  measured,  even  if  the  instant  of  cul- 
mination be  accurately  noted  by  a  chronometer.  But  if  Polaris  be 
the  star  chosen,  a  series  of  observations  may  be  made  during  the 
five  minutes  immediately  preceding  and  following  culmination,  and 
at  no  time  during  these  ten  minutes  will  the  star's  altitude  differ 
from  its  meridian  altitude  by  more  than  I'M.  Errors  within  this 
limit  would  not  be  detected  by  even  the  best  sextant  observations, 
and  the  mean  of  the  measured  altitudes  will  therefore  be  the  me- 
ridian altitude  with  the  usual  precision. 

Even  if  a  be  regarded  as  too  small  when  found  in  this  manner, 
a'  will  be  too  large  by  practically  the  same  amount,  and  £  (a  -j-  a') 
will  be  correct. 

2.  By  Meridian  Altitudes  or  Zenith  Distances. — This  method  de- 
pends on  the  fact  that  the  astronomical  latitude  of  a  place  is  equal 


LATITUDE.  79 

to  the  declination  of  its  zenith.    If  the  star  culminate  between  the 
pole  and  the  zenith,  then 

0=0-3,, 

where  Z^  is  the  meridian  zenith  distance  of  the  star.     If  between 
the  zenith  and  equator,  then 


We  have  therefore  only  to  measure  zl  ,  take  d  from  the  Ephemeris., 
and  substitute  in  one  of  these  equations. 

This  method  is  a  very  exact  one  when  the  observations  are  made 
with  an  instrument,  such  as  the  transit  circle,  accurately  adjusted 
to  the  meridian,  and  whose  least  count  is  small.  It  is  subject  to 
errors  of  both  declination  and  refraction;  although  the  latter  as 
well  as  any  constant  errors  in  the  measured  altitudes  may  be  nearly 
eliminated,  as  is  seen  from  the  preceding  equations,  by  combining 
the  result  with  that  from  another  star  which  culminates  at  about 
the  same  time  at  a  nearly  equal  altitude  on  the  opposite  side  of  the 
zenith. 

For  reasons  stated  above,  the  sextant  is  not  well  adapted  to  thi8 
method  except  at  sea,  where  the  highest  accuracy  is  not  requisite. 

3.  By  Circum-meridian  Altitudes.  —  If  the  altitude  of  a  celestial 
body  be  measured  within  a  few  minutes  of  culmination,  we  may  by 
noting  the  corresponding  time  very  readily  compute  the  difference 
between  the  measured  altitude  and  the  altitude  which  the  body 
will  have  when  it  reaches  the  meridian.  This  difference  is  called 
the  "  Keduction  to  the  Meridian,"  and  by  addition  to  the  observed 
will  give  the  meridian  altitude.  If  several  altitudes  be  measured 
and  each  be  reduced  to  the  meridian,  we  may  evidently,  by  taking 
the  mean  of  the  results,  obviate  the  inaccuracies  incident  to  the 
use  of  the  sextant  in  the  last  problem. 

These  are  called  "  Circum-meridian  Altitudes,"  and  their  reduc* 
tion  to  the  meridian  is  rendered  very  simple  by  the  special  formula 

cos  0  cos  d    2  sin2  ^  P 


cos  at  sin  1" 

P 


'  ~ 

/cos  0  cos  6\\          2  sin4  1 
—    -  —  I  tan  a.  •  —  :  —  ^T.  --  h 

\     cos  at     )  '     sm  1" 

the  deduction  of  which  will  be  given  hereafter. 


SO  PRACTICAL  ASTRONOMY. 

In  this  formula  a  is  the  true  altitude,  d  the  declination,  and  P 
the  hour  angle,  all  relating  to  the  instant  of  observation ;  a/  is  the 
desired  meridian  altitude,  and  the  second  and  third  terms  of  the 
second  member  constitute  the  first  two  terms  of  the  Reduction  to 

77      __    ....          -- ,  .  2  sin2 -I- P        .2  sin*  IP 

the  Meridian.     Values  of  , — : — -~  and   • — : — —-=—  are  given  in 

sin  1"  sin  1" 

tables  with  P  as  the  argument.  For  small  values  of  P  the  series 
will  converge  rapidly,  provided  af  is  not  too  large. 

Having  the  meridian  altitude,  the  latitude  follows  as  in  the  last 
method. 

From  (128)  it  is  seen  that  for  computing  at  we  require  (neg- 
lecting all  consideration  of  P  for  the  present)  not  only  d,  but  both 
at  and  0 ;  but  as  will  appear  later,  approximate  values  will  suffice. 
If  an  approximate  value  of  0  be  known,  that  of  a,  follows  from 

at  =  d  +  90°  -  0.  (129) 

If  not,  one  may  be  found  as  follows :  In  this  method,  double  altitudes 
are  taken  in  as  quick  succession  as  possible  from  a  few  minutes  before 
until  a  few  minutes  after  meridian  passage.  The  greatest  altitude 
measured  will  therefore, when  corrected  for  refraction,  semi-diameter, 
and  parallax,  be  very  near  the  meridian  altitude,  and  its  substitution 
in  (129)  will  give  a  value  of  0  sufficiently  accurate  for  the  purpose. 

In  order  to  fix  upon  a  proper' value  of  d  to  be  used  in  (128)  it 
is  to  be  noted  that  if  a  star  be  the  body  observed,  its  declination  is 
practically  constant  and  may  be  taken  at  once  from  the  Ephemeris 
for  the  date.  In  case  of  the  sun,  however,  whose  declination  is 
constantly  varying,  d  must  represent  the  declination  at  the  moment 
of  making  the  observation.  But  when  several  observations  are 
taken  in  succession,  the  labor  of  computing  a  value  of  d  for  each 
may  be  avoided,  as  will  be  evident  from  an  explanation  of  the 
manner  of  making  the  observations  and  reductions. 

The  observations  ar'e  made  as  just  explained  on  a  limb  of  the 
sun,  viz. :  Several  double  altitudes  are  taken  as  near  together  as 
possible,  as  many  before,  as  after  meridian  passage,  and  the  corre- 
sponding chronometer  times  noted.  (Note  the  difference  between 
this,  and  sextant  observations  for  time.) 

Now  if  we  suppose  each  observation  to  have  been  reduced  to 
ihe  meridian,  after  correcting  for  refraction,  parallax  and  semi- 
diameter,  we  would  have  several  equations  of  the  form 

at  =  a  -f-  A  m  —  B  n, 


LATITUDE.  81 

„  2  sin2  \P       _  2  sin4  4  P 

in  which  m  and  ^  are  the  tabular  values  of  — . — 7,7—,  and  — r — 77—, 

sm  1"  sin  1" 

md  ^4  and  B  the  remaining  factors  of  the  corresponding  terms  in 
Equation  (128).  Any  one  of  the  equations  will  give  for  the  lati- 
tude, 

0  =  3  +  90°  -  (a  +  A  m  -Bri).  (130) 

In  this  equation,  &  is  the  declination  at  the  time  of  observation. 
For,  since  the  reduction  to  the  meridian  has  been  made  with  this 
value  of  6  in  obtaining  A  and  B,  a  ~f-  A  m  —  B  n  is  manifestly  the 
meridian  altitude  of  a  body  whose  declination  is  constantly  d.  In 
fact,  the  reduction  to  the  meridian  by  the  formula  given,  can  be 
computed  only  on  the  hypothesis  of  a  constant  declination.  We 
are  thus  dealing  with  a  fictitious  sun,  whose  declination  on  the  me- 
ridian differs  from  that  of  the  true  sun.  But  since  declination  and 
meridian  altitude  always  preserve  a  constant  difference  (the  colati- 
tude),  we  see  that  Equation  (130)  will  give  the  correct  value  of  0, 
due  to  perfect  balance  in  the  errors  of  tf  and  (a  -\-  A  m  —  B  n). 

The  mean  of  all  the  equations  due  to  the  several  observations 
will  be 

0  =  <*0  +  90°  -  K  +  A0m0  -  S0n0).  (131) 

In  this  equation  6\  is  the  mean  of  the  sun's  declinations  at  the 
times  of  making  the  observations;  and  it  is  obvious  that  if  this 
mean  be  employed  for  the  single  computation  of  A0  and  B0 ,  the 
error  committed  will  be  entirely  negligible.  We  thus  avoid  a 
separate  computation  of  these  quantities  for  each  observation. 

The  result  will  moreover  be  perfectly  rigorous  in  practice  if  we 
use  for  <50  the  declination  corresponding  to  the  mean  of  the  times; 
since  in  the  30  minutes  covered  by  the  observations  the  departure 
of  the  sun's  declination  from  a  uniform  increase  or  decrease  is 
negligible.  We  thus  avoid  the  labor  of  computing  more  than  a 
single  value  of  tf. 

We  have  still  to  determine  the  value  of  P  from  the  chronometer 
time  of  each  observation,  and  in  this  determination  it  must  be 
borne  in  mind  that  P  (in  arc)  is  the  angular  distance  of  the  true 
sun  from  the  meridian  at  the  instant  of  observation. 


82  PRACTICAL  ASTRONOMY. 

There  are  two  reasons  why  this  distance  (in  time)  cannot  be 
given  directly  by  a  mean  time  chronometer.  First,  the  chronom- 
eter will  usually  be  gaining  or  losing,  i.e.,  it  will  have  a  "  rate" 
Secondly,  a  mean  time  chronometer,  even  when  running  without 
rate,  indicates  the  angular  motion  of  the  mean  sun,  which  may  be 
quite  different  from  that  of  .the  true^siin,  as  shown  by  the  continual 
change  in  the  Equation  of  Time. 

We  therefore  proceed  as  follows:  From  Page  I,  Monthly  Calen° 
dar  of  the  Ephemeris  (knowing  the  longitude),  take  out  the  Equa- 
tion of  Time.  Add  this  algebraically  to  12  hours,  apply  the  error 
of  the  chronometer,  and  the  result  will  be  the  chronometer  time 
of  apparent  noon.  The  difference  between  this  and  the  chro- 
nometer time  of  each  observation,  gives  the  several  values  of  P  in 
time,  each  subject  to  the  two  corrections  mentioned.  To  find  the 
correction  for  rate,  let  r  represent  the  number  of  seconds  gained  or 
lost  in  24  hours  (a  losing  rate  being  positive  for  the  same  reason 
that  an  error  slow  is  positive).  Then  if  P'  be  the  corrected  hour 
angle,  we  will  have 

P'  :  P  ::  86400  :  86400  -  r.        [86400  =  60  X  60  X  24]. 
Or 

86400 
86400  -  r 

Or 

2j3in9J-P'_  2  sin2  j  P  /     86400    \*  _     2  sina  j  P 
sin  1"  sin  1"    \86400  —  r)   '~~   "     sin  1"~ 

•/- 

Hence  we  will  also  have 

cos  0  cos  d  2  sin2  i  P 


A      i  L  j  *         ,  \       7 

Am  (corrected  tor  rate)  =  k 


Hence  if  we  compute  A  by  the  formula 


__  -,  cos  0  cos 
—  Ic 


sin 


LATITUDE.  83 

we  may  employ  the  actual  chronometer  intervals  and  pay  no  fur- 
ther attention  to  the  question  of  rate. 

From  k  =  (  —        — ]  ,  values  of  k  are  tabulated  with  the  rate 
\oo40U  —  TI 

as  the  argument. 

The  second  correction  depends,  as  just  stated,  on  the  difference 
between  the  motions  of  the  true  and  mean  sun,  while  the  former  is 
passing  from  the  point  of  observation  to  the  meridian.  In  other 
words  it  depends  on  the  change  in  the  Equation  of  Timn  in  the 
•same  interval,  or,  which  is  the  same  thing,  upon  the  rate  ff  an  ac- 
curate mean  solar  chronometer  on  apparent  time. 

If  therefore  we  let  e  represent  the  change  in  the  Equ.ition  of 
Time  for  24  hours  (positive  when  the  Equation  of  Timo  ic  increas- 
ing algebraically),  it  is  evident  that  r  —  e  will  be  the  rat}  of  the 
given  chronometer  on  apparent  time,  and  that  the  correction  for 
this  total  rate  may  be  computed  as  just  explained  for  r,  or  taken 
from  the  same  table,  using  r  —  e  as  the  argument  instead  of  r  alone. 

The  operation  of  reducing  the  observations  is  then,  in  )»rief,  as 
follows. 

By  Circum- Meridian  Altitudes  of  the  Sun's  Limb. — Fo^m  7. 

Correct  the  mean  of  the  double  altitudes  for  eccentricity  and  in- 
dex error.  Correct  the  resulting  mean  single  altitude  for  refraction, 
semi-diameter,  and  parallax  in  altitude.  Denote  the  result  by  a0. 

From  the  Equation  of  Time  (Page  I,  Monthly  Calendar),  longi- 
tude and  chronometer  error,  find  the  chronometer  time  of  apparent 
noon. 

Take  the  difference  between  this  and  each  chronometer  time  of 
observation,  denote  the  difference  by  P,  and  their  mean  by  Pn. 

With  each  value  of  P,  take  from  tables  the  corresponding 
values  of  m  and  n.  Denote  their  respective  means  by  m0  and'  n0. 

From  Page  II,  Monthly  Calendar,  take  the  sun's  declination 
corresponding  to  the  local  apparent  time  P0,  and  denote  it  by  tffl. 

If  0  can  be  assumed  with  considerable  accuracy,  determine  the 
corresponding  at  by  «,  =  #„  +  90°  —  0. 

If  not,  take  the  greatest  measured  altitude,  correct  it  for  refrac- 
tion, etc.,  call  it  a, ,  and  deduce  0  from  the  above  equation. 

From  the  rate  of  the  chronometer  and  change  in  Equation  o^ 
Time,  (both  for  24  hours,)  take  k  from  the  table. 


84  PRACTICAL  ASTRONOMY. 

With  these  values  of  Jc9  </>,  at ,  and  #0 ,  compute 

cos  0  cos  tf0 

^0  =  -  -  Jc,    and    B0  =  A*  tan  dL. 

cos  at 

The  latitude  then  follows  from 

0  =  <J0  +  90°  -  K  +  Aum0  -  BjiJ.  (132) 

.##  Circum-Meridian  altitudes  of  a  Star. — Form  8. 

With  a  star  observed  with  a  sidereal  chronometer,  the  observa- 
tions are  the  same,  and  the  reduction  is  only  modified  by  the  fact 
that  parallax,  semi-diameter,  equation  of  time  and  longitude  do  not 
enter,  while  the  declination  is  constant. 

If  the  star  lie  between  the  zenith  and  pole,  the  formula  becomes 

0  =  («0  +  A0m0  -  B0n0)  -  90°  +  <*0.  (133) 

If  below  the  pole, 

0  =  K  -  A0mQ  -  £0n0)  +  90°  -  *0.  (134) 

1.  An  Ephemeris  star  is  to  be  preferred  to  the  sun,  since  the 
reduction  is  more  simple,  its  declination  is  better  known  and  con- 
stant, it  presents  itself  as  a  point,  which  is  of  advantage  in  sextant 
observations,  and  we  have  a  greater  choice  both  in  time  and  the 
place  of  the  object  to  be  observed. 

2.  By  comparing  Eqs.  (132)  and  (133)  we  see  that  constant 
errors  in  the  measured  altitudes,  and  in  refraction,  will  be  nearly 
eliminated   by  combining  the  results  of  two  stars,  one  as  much 
north  as  the  other  is  south,  of  the  zenith. 

Also  from  the  principal  term  of  the  Reduction  to  the  Meridian, 

cos  (b  cos  §     2  sin2  4-  P   .  /v       ,1 

-  .  — : j-. — ,  it  is  seen  that  the  effect  of  an  imperfect 

cos  at  sin  1 

knowledge  of  the  chronometer  error,  giving  an  incorrect  valuj  of 
P  may  be  eliminated  by  taking  another  observation  at  about  an 
equal  altitude  on  the  other  side  of  the  meridian  ;  since,  P  being 
very  small,  sin2  \  P  will  be  as  much  too  large  in  one  case  as  it  will 
be  too  small  in  the  other. 


LATITUDE.  85 

The  double  altitudes  should  therefore  be  taken  at  as  nearly 
equal  intervals  of  time  and  be  as  symmetrically  arranged  with  refer- 
ence to  the  meridian,  as  practicable. 

3.  By  rewriting  the  assumed  formula  for  the  reduction,  ex 
pressing  the  first  term  as  a  function  of  0  and  8  only,  and  including 
the  third  term  which  has  heretofore  been  omitted,  we  have  (Form- 
ula 2,  P.  4,  Book  of  Formulas), 


______! _  tan  a,  |  (1  -{-  3  tan2  at) 

'          ~tan0-tanrf    '       (tan  0  -  tan  tf)2  *  h  (tan0-tan^)3^ 

and 

_  1  tan  a,  f  (1  +  3  tan2 «,) 

^'  "        ~  tane?-tan0™  ~  (tan  d  -  tan  0)2  n  *"  (tan  tf  -  tan  0)A 

for  south  and  north  stars  respectively. 

P      _  2  sin2  j  P          _  2  sin4  -|  P          _  2  sin8  j  P"[ 
L^         sin  1"   '  sin  1"    '  sin  1"   J" 

From  these  equations  it  is  seen  that  if  a  star  be  selected  which 
culminates  at  a  considerable  distance  from  the  zenith,  either  north 
or  south,  the  first  factor  of  each  term  of  this  development  is  much 
smaller  than  in  case  of  a  star  culminating  near  the  zenith,  either 
north  or  south. 

Since  the  third  term  has  been  entirely  neglected  in  the  previous 
discussion,  it  becomes  desirable  to  select  our  star  in  such  a  manner 
that  the  omitted  term  (and  hence  all  following  it)  shall  be  small; 
and  this,  as  just  seen,  will  occur  when  there  is  considerable  differ- 
ence between  the  latitude  and  the  star's  declination  in  either  direc- 
tion. It  is  also  seen  that  an  unfavorable  position  of  the  star  near 
the  zenith  causing  the  first  factor  to  be  excessive  may  be  counter- 
balanced by  diminishing  the  hour  angle  P. 

From  the  above  expression  for  the  third  term,  knowing  the  ap- 
proximate latitude,  we  may  readily  find  the  hour  angle  of  any  given 
star,  within  which  if  the  observation  be  confined,  the  value  of  the 


86  PRACTICAL  ASTRONOMY. 

term  will  not  exceed  any  desired  limit — say  0".01  or  1".  Similarly 
for  the  second  term.  We  thus  ascertain  how  long  before  culmina- 
tion the  observations  may  safely  be  begun  when  it  is  proposed  to 
omit  one  or  both  terms  in  the  reduction. 

For  example  in  latitude  40°  N.,  if  we  observe  a  star  at  declina- 
tion 0°.  the  observation  may  be  made  at  20m  from  meridian  passage 
and  yet  the  third  term  amount  only  to  .01",  which  would  affect  the 
resulting  latitude  by  one  linear  foot.  Or  it  may  be  made  at  27in 
from  culmination,  and  the  third  term  amount  only  to  .1",  affecting 
the  resulting  latitude  by  ten  feet. 

A  star  at  an  equal  altitude  north  of  the  zenith,  declination  80° 
(for  combination  with  the  preceding  as  recommended),  may  be 
observed  at  48  and  62  minutes  from  culmination,  with  no  larger 
errors. 

With  other  latitudes  the  figures  will  vary,  but  the  principle  re- 
mains the  same. 

Hence  the  general  rule :  Select  a  star  whose  declination  differs 
considerably  from  the  latitude.  This  will  give  ample  time  for  tak- 
ing a  series  of  altitudes.  As  the  declination  of  the  selected  star 
approaches  the  latitude,  restrict  the  observations  to  a  shorter  time, 
greater  care  in  this  respect  being  necessary  for  south  stars.  Ar- 
range the  observations  as  symmetrically  ivith  reference  to  the  me- 
ridian as  practicable,  and  use  at  least  two  stars — on  opposite  sides 
of  the  zenith. 

4.  Finally,  if  a  mean  solar  chronometer  be  used  with  a  star,  the 

corrected  m.  s.  intervals  defined  by  the  equation  P'  =  P  --- 


80400  -r 

must  evidently  be  reduced  to  sidereal  intervals  by  multiplying  by 
1.00272791  heretofore  deduced.     That  is 


and  the  factor  for  rate  will  be  k  (1.00273791)3  instead  of  k. 

Similarly  if  a  sidereal  chronometer  be  used  with  the  sun, 
factor  for  rate  will  be  k  (0.99726957)2  instead  of  k. 


LATITUDE.  87 

5.  To  Determine  the  Reduction  to  the  Meridian.—  The  difference 
between  a  circum-meridian  and  the  meridian  altitude  of  a  body  is 
called  the  "  Reduction  to  the  Meridian." 

Its  nature  will  be  understood  from  Figure  18.  S  is  the  place 
of  the  star;  S'  the  point  where  it 
crosses  the  meridian,  (P  S  =  P  S')  ; 
and  SS"  the  arc  of  a  small  circle 
of  which  Z,  the  zenith,  is  the  pole, 
(ZS"  =  ZS).  Z  S'  will  therefore 
be  the  meridian  zenith  distance 
=  *,  =  90°  -  at\  ZS  or  Z  S"  will 
be  the  circum-meridian  zenith  dis-  FlG*  18> 

tance  =  z  =  90°  —  a\  and  if  x  denote  the  Reduction  to  the  Meri- 
dian =  S'  8",  we  shall  have 


The  several  terms  of  Equation  (128)  after  a  therefore  represent 
x;  and  it  is  required  to  deduce  this  value  of  x  arranged,  as  is  seen, 
in  a  series  according  to  the  ascending  powers  of  sin2  |  P, 

The  equation  heretofore  deduced,  viz.  :  ' 

cos  z  =  sin  0  sin  #  -\-  cos  0  cos  d  —  2  cos  0  cos  $  sin2  £  P, 

gives  by  reduction  (since  sin  0  sin  6  -f  cos  0cos  d  =  cos  (0  —  tf) 
=  cos  zt), 

cos  zt  —  cos  z  —  2  cos  0  cos  d  sin2  \  P  =  0.  (  £) 

Putting  for  convenience  2  cos  0  cos  #  =  m,  and  sin2  £  P  =  y, 
we  have 


cos  zt  —  cos  z  —  my  =  0. 
We  also  have 

+  ^,  (c) 


cos  2  =  cos  x  cos  3,  —  sin  x  sin  z,. 
Hence  from  (bf) 

cos  2,  —  cos  x  cos  2,  -j-  sin  #  sin  2/  —  m  y  —  0.  (d) 

Now  let 


88  PRACTICAL  ASTRONOMY. 

be  the  undetermined  development  desired.  From  the  relation  ex- 
pressed by  (d),  we  are  to  determine  such  constant  values  of  A,  B, 
and  C,  as  will  make  the  series,  when  convergent,  true  for  all  values 
of  y.  Therefore  let  the  values  of  cos  x  and  sin  x  derived  from  (e) 
be  substituted  in  (d).  The  resulting  equation  will,  from  the  con- 
dition imposed  on  (e),  be  an  identical-'equation. 

To  find  cos  x  and  sin  x  for  this  substitution,  we  have  from  cal- 
culus, 

x* 

cos  x  =  1  —  —  +  etc., 
4 

-  1~  • 

x3 
sin  x  =  x  —  —  -f-  etc., 

and  from  (e), 

cos  x  =  1  -  i  (AY  +  2  A  B  if  -f  etc.), 
sin  x  =  A  y  -f  B  y*  -f  Cy*  —  \  A3y3  —  etc. 

Substituting  in  (d),         . 

cos  zt  —  cos  zt  -f  \  A1  cos  2,  ?/2  -f  ^4  5  cos  2,  y3  -f  sin  z, 
-f-  sin  ztB  y1  -\-  sin  2,  (7?/3  —  ^-  sin  zt  A3y3  —  m  y  —  0. 

Collecting  the  terms, 

\ 

cos  z 


. 
sm  z.A  )       .    (  A  A9  cos  z.  )    a  .    \    .     .        n      (    *      f\ 

!•  y  +  •!  ,  T>  -      >  v  +  K  +  sm  2.  (7     hir.r*  °- 

—  m  \  y    r  (  +  B  sins,    ^       /    '        .  '       .3  \  ^ 

(  ~  |-  sin  2/  ^4  ; 

From  the  principles  of  identical  equations 


sin  zt  A  —  m  =  0.     A  =  - 


m 


1  m  cos  z.  .    „    .  „  1m   cot  2/ 

-  —  ^-5  —  '  +  B  sm  2;  .  —  0.     B  —  —  -  —  ^-5  —  ' 
sin2  2  2     sma  z 


cot2  z.    .   1     m3          .         „  „      1 


.         „  „ 

sm^.C'^O.     (7=  -   . 
' 


-  —  ^-^  —         *  ..  . 

2      sm2  zt          6  sin2  2,  '  6  sm 


THE  ZENITH  TELESCOPE.  89 

Therefore 

cos  0  cos  d n    .     1  /cos  0  cos  #  \3  .     1 

3  =  -  —  2sma-P  —    -  -    tan  a,  2  sm4  -P 

cos  a/  2  V      cos  a       I  2 


cos 


Reducing  the  terms  of  the  series  from  radians  to  seconds  of  arc, 
we  have  for  the  value  of  at  , 

,   cos  0  cos  d  2  sin2  4-  P        /cos  0  cos  tf\2  ^  2  sin4  4  P 

a  —  a  -f  -  —  —  _i-  --    -  -     tan  a.  —  .    ,*, 

cos  a,          sm  1"  \     cos  at      )  sm  1" 

,  2  /cos  0  cos  <y\"  x  _  sin6  A  P 

+  TT  -  -  H1  +  3tan  <O  %     •    \ff  —  etc. 

1   3  \      cos  a       J  v  sm  1" 


THE  ZENITH   TELESCOPE. 

This  instrument,  being  employed  in  the  next  latitude  problem, 
will  now  be  briefly  described,  and  the  manner  of  determining  its 
constants  explained.  Its  use,  as  will  be  seen,  is  limited  to  field 
work,  and  it  therefore  forms  no  essential  part  of  the  equipment  of 
a  permanent  observatory. 

The  instrument  consists  of  a  telescope  like  that  of  the  transit, 
mounted  at  one  end  of  a  horizontal  axis,  counterpoised  by  a  weight 
at  the  other.  The  telescope  turns  freely  in  altitude  about  this  axis, 
which  is  in  turn  supported  by  a  conical  vertical  column  rising  from 
the  centre  of  a  horizontal  graduated  circle,  the  circle  resting  on  a 
small  frame  consisting  of  three  legs  whose  feet  are  levelling  screws. 

The  horizontal  axis  with  the  telescope  attached  turns  freely  in 
azimuth  about  the  vertical  column,  the  amount  of  such  motion 
being  indicated  by  a  vernier  sweeping  over  the  horizontal  circle. 
By  this  motion  the  instrument  is  placed  in  the  meridian. 

The  setting  circle  is  similar  to  the  one  described  in  connection 
with  the  transit.  It  is  rigidly  attached  to  the  body  of  the  telescope, 
and  reads  to  single  minutes  of  zenith  distance.  The  attached  level, 
connected  with  the  movable  vernier  arm  of  the  setting  circle,  being 
intended  to  measure  as  well  as  to  indicate  differences  of  mclina- 


PRACTICAL  ASTRONOMY, 


FIG,  19. —THE  ZENITH  TELESCOPE, 


THE  ZENITH  TELESCOPE. 


91 


tion,  is  of  considerable  delicacy.  The  instrument  is  provided  with 
cl-amp  and  tangent  screws  for  both  motions,  also  the  usual  adjusting 
screws. 

The  field  of  view  presents  the  appearance  shown  in  Figure  20; 
sometimes  however  the  number  of 
vertical  wires  is  increased  so  that 
the  instrument  may  if  necessary  be 
used  as  a  transit.  The  wires  are 
all  fixed  except  i  k,  which  can  be 
moved  up  or  down  parallel  to  it- 
self, and  is  called  the  declination 
micrometer  wire.  The  comb-scale 
f  g  is  so  cut  that  one  turn  of  the 
micrometer  head  carries  the  wire 
i  k  exactly  from  one  tooth  to  the 
next,  thus  recording  the  number 
of  whole  revolutions  between  two 
positions  of  the  wire.  Hundredths  of  a  revolution  are  shown  on  the 
micrometer  head  by  a  fixed  index.  These  are  called  divisions. 
(Arrangements  for  illuminating  the  wires  are  the  same  as  with  the 
transit.) 

Therefore  it  is  seen  that  if,  when  the  instrument  is  adjusted  to 
the  meridian,  two  stars  cross  the  middle  wire  at  different  times  and 
in  different  places,  but  yet  within  the  same  field  of  view,  we  may 
find  the  difference  of  their  meridian  zenith  distances  by  bisecting 
each  in  succession  by  the  movable  at  the  instant  of  its  passing  the 
middle  wire,  noting  the  difference  of  micrometer  readings,  and 
multiplying  the  result  by  the  value  in  arc  of  one  division  of  the 
micrometer  head :  or  if  the  attached  level  shows  the  telescope  to 
have  altered  its  angle  of  elevation  between  the  observations,  thus 
apparently  displacing  the  second  star  in  the  field  of  view,  we  may 
still  correct  the  micrometer  reading  provided  we  know  the  value  in 
arc  of  one  division  of  the  level. 

It  is  therefore  necessary  to  determine  for  each  instrument  these 
two  constants. 

The  Attached  Level  and  Declination  Micrometer  of  the  Zenith 
Telescope. — Since  the  level  is  neither  detached,  nor  attached  to  a 
circle  reading  to  seconds,  neither  of  the  modes  of  finding  the  level 
constant  given  in  connection  with  the  transit,  is  available.  The 


92  PRACTICAL  ASTRONOMY. 

same  is  true  of  the  micrometer  constant,  since  the  micrometer  wire 
is  now  parallel,  not  perpendicular,  to  the  apparent  path  of  a  star  at 
its  meridian  passage.  With  the  zenith  telescope  the  usual  method 
of  finding  these  two  constants  is  to  find  the  value  of  a  division  of 
the  level  in  terms  of  a  revolution  of  the  micrometer  head.  Then 
after  finding  the  latter  (which  Wolves  the  former)  we  may  find 
the  actual  value  of  a  division  of  the  level  in  seconds  of  arc,  as  will 
now  be  explained.  The  formulas  come  at  once  from  the  astronom- 
ical triangle,  remembering  that  ac  the  time  of  a  star's  elongation 
the  triangle  is  right-angled:  (ip  =  90°). 

Direct  the  telescope  to  a  small,  well-defined,  distant,  terrestrial 
object,  and  set  the  level  so  that  the  two  ends  of  the  bubble  will  give 
different  readings.  Bisect  the  object  with  the  micrometer  wire, 
note  the  reading,  also  that  of  each  end  of  the  bubble.  Mo?e  the 
telescope  and  level  together  by  the  tangent  screw  until  the  bubble 
plays  near  the  other  end  of  the  tube.  Again  bisect  the  mark  by  the 
micrometer  wire  and  note  all  three  readings  as  before.  The  mean 
of  the  number  of  divisions  passed  over  by  the  two  ends  of  the  bub- 
ble is  then  the  number  of  divisions  passed  over  by  the  bubble.  The 
difference  of  the  micrometer  readings  is  the  run  of  the  micrometer. 
Dividing  the  second  by  the  first,  we  have  the  value  of  a  division  of 
the  level  in  terms  of  a  revolution  of  the  micrometer.  Take  a  mean 
of  several  determinations  and  denote  it  by  d. 

We  can  now  find  the  value  of  one  division  of  the  micrometer. 
For  reasons  stated  when  treating  of  the  R.  A.  micrometer,  we  use  a 
circumpolar  star,  and  at  the  instant  that  its  path  is  perpendicular 
to  the  wire  in  question.  This  requires  us  to  take  the  star  at  its 
elongation.  Manifestly  the  same  principles  apply  to  the  two  cases, 
since  the  principal  difference  is  that  the  star  and  wire  have  each 
been  apparently  shifted  90°;  the  motion  of  the  star  with  reference 
to  the  wire  not  having  changed.  Some  changes  in  detail  are  how- 
ever necessary.  In  the  first  place,  since  the  motion  of  the  star  is 
almost  wholly  in  altitude,  we  cannot  as  before  neglect  differences  in 
refraction  between  two  transits.  Again,  since  the  pressure  of  the 
hand  in  working  the  micrometer  head  is  in  a  direction  to  cause  a 
possible  disturbance  of  the  instrument  even  though  firmly  clamped, 
we  must  read  the  level  at  every  transit,  and  if  any  change  has  oc- 
curred, correct  the  micrometer  readings  accordingly. 

As  a  preliminary,  we  must  determine  the  time  of  elongation 


THE  ZENITH  TELESCOPE.  93 

(in  order  to  know  when  to  begin  our  observations),  and  the  setting 
of  the  instrument,  i.e.,  the  azimuth  and  zenith  distance  of  the  star 
at  the  time  of  elongation.  The  hour  angle  is  found  from 

cos  P0  =  cot  3  tan  0, 
from  which  the  sidereal  time  of  elongation  is  given  by 

f~t~f  ,,       t         T">  XT  /  "I  O  f  \ 

./o  =  a  ±  Jr0  —  A  (135) 

in  which  a  is  the  star's  apparent  R.  A.  for  the  instant,  and  E  is  the 
error  of  the  chronometer.     The  plus  sign  is  used  for  western  and 
the  minus  for  eastern  elongations. 
The  azimuth  is  given  by 

sin  A  =  ^-7,  (136) 

COS0 

and  the  zenith  distance  by 

sin0  .       N 

cos  Z0  =•  — — — .  (lo  7 ) 

Set  the  instrument  in  accordance  with  these  coordinates  20  or 
30  minutes  before  the  time  of  elongation,  and  as  soon  as  the  star 
enters  the  field,  shift  the  telescope  if  necessary  so  that  it  will  pass 
nearly  through  the  center. 

The  observations  are  now  conducted  in  exactly  the  same  man- 
ner as  for  the  R.  A.  micrometer,  with  the  addition  that  each  end  of 
the  level  bubble  is  read  in  connection  with  each  transit. 

Then,  as  before,  each  observation  is  compared  with  the  one 
made  nearest  the  time  of  elongation,  T0,  the  interval  of  time 
being  computed  from  either 

sin  i  =  sin  /cos  d,  (137J) 

or 

i  =  /cos  #, 

i  • 

according  to  the  declination  of  the  star.  After  which  we  have  in 
arc  (neglecting  for  the  present  differences  of  refraction  and  level), 

15* 


94  PRACTICAL  ASTRONOMY. 

M  being  the  number  of  micrometer  revolutions  or  divisions  be- 
tween the  two  positions  of  the  star,  and  R'  the  value  of  one  revolu- 
tion or  division. 

But  if  the  reading  of  the  level  is  different  at  the  two  observa- 
tions, manifestly  Mmust  be  corrected  accordingly. 

For  instance,  if  the  level  shows  that  between  the  two  observa- 
tions the  telescope  had  moved  with  the  strr  in  its  diurnal  path, 
then  evidently  the  micrometer  will  indicate  only  a  part  of  the 
angular  distance  between  the  two  positions  of  the  star,  and  the 
level  correction  must  be  added  to  the  micrometer  interval.  Con- 
versely, if  the  telescope  has  moved  against  the  motion  of  the  star. 
This  level  correction  is  found  as  follows  :  if  d  is  the  value  of  one 
division  of  the  level  in  terms  of  a  revolution  of  the  micrometer, 
and  L  the  number  of  divisions  which  the  level  has  shifted,  then 
Ld  will  be  the  value  (in  micrometer  revolutions)  of  the  correction 
to  be  applied  to  M.  The  method  of  finding  d  has  already  been 
explained. 

Hence  the  value  of  R'  becomes, 


M±L# 

Since,  however,  refraction  affects  the  two  positions  of  the  star  un- 
equally, it  is  seen  that  M  ±  L  d  is  only  the  difference  of  apparent 
zenith  distances  (i.e.,  the  instrumental  difference),  while  15  i  being 
derived  directly  from  the  time  interval,  is  the  difference  of  true 
zenith  distances.  If  therefore  15  i  be  corrected  by  the  difference 
of  refraction,  the  numerator  will  denote  the  difference  of  apparent 
zenith  distance  in  arc,  and  the  denominator  this  same  difference  in 
micrometer  revolutions. 

Denote  by  A  r  the  difference  of  refraction  in  seconds  for  1'  of 
zenith  distance  at  z0;  then  for  15  i"  it  may  be  taken  as  -^  15  i  A  /*, 
which  is  the  desired  correction.  The  above  formula  ,  therefore  be- 
comes, denoting  the  true  value  of  a  revolution  by  R, 

R'Ar . 
K S7T~>  Vus) 


M±Ld  60 

A  r  is  taken  from  refraction  tables. 


THE  ZENITH  TELESCOPE.  95 

i 

The  adopted  value  of  R  should  be  a  mean  of  the  results  from 
all  the  observations. 

Having  now  found  R,  the  value  in  arc  of  one  division  of  the 
level  is  evidently 

D  =  Rd,  (139) 

since  d  is  the  value  in  micrometer  revolutions.     Both  constants  are 
therefore  determined. 

One  of  the  most  convenient  and  accurate  modes  of  employing 
formula  (138)  in  practice,  is  as  follows  :  Suppose  the  star  to  be 
approaching  eastern  elongation,  and  the  micrometer  readings  to 
increase  as  the  zenith  distance  decreases.  Let  Z0  ,  M0  ,  and  L0  be 
the  zenith  distance,  micrometer,  and  level  readings  at  elongation 
(all  unknown),  and  Z',  M',  and  L'  the  corresponding  quantities  at 
the  time  of  any  one  of  the  recorded  transits.  Then  remembering 
that  in  (138),  15  i  is  the  true  difference  of  zenith  distance 
=  Zf  —  Z^  ,  M—  M0  —  Mf,  L  =  L0  —  L',  and  reserving  the  correc-. 
tion  for  refraction  to  be  applied  finally,  we  have 

Z'-ZQ=  (M0-M')  £  +  (£.-  L')Rd. 

Similarly  for  another  transit, 

Z"  -ZQ  =  (M0  -  M")  R  +  (L0-  L")  R  d. 
Subtracting  and  solving, 


(M"  -  M')  +  (L"  -  L')d' 

Then  Z  —  Z0  having  been  computed  for  each  transit  by 
these  differences  may  be  taken  by  pairs  for  substitution  in  (140),  in 
any  manner  desired.  For  example,  if  forty  transits  have  been  re- 
corded, it  is  usual  to  pair  the  first  difference  with  the  twenty-first, 
the  second  with  the  twenty-second,  etc.,  when  if  the  successive 
micrometer  readings  have  been  equidistant,  the  divisors  will  be 
equal,  save  for  the  slight  level  correction.  We  thus  obtain  twenty 
determinations  of  R,  the  mean  of  which  should  be  corrected  for 

/?  A  T 

refraction  as  shown  in  (138),  viz.  :  by  subtracting  . 

uO 


96  PRACTICAL  ASTRONOMY. 

The  preceding  method  of  finding  these  two  constants  of  the 
zenith  telescope  is  regarded  as  the  best;  but  provision  is  made  in 
the  construction  of  the  instrument  for  turning  the  box  containing 
the  wire  frame  thicugh  an  angle  of  90°.  When  this  is  done,  the 
declination  micrometer  becomes  virtually  a  K.  A.  micrometer,  and 
the  value  of  a  revolution  may  be  found  as  described  for  that  mi- 
crometer, and  then  the  box  revolved  back  to  its  proper  place  and 
clamped.  In  this  case  however  the  result  must  be  in  arc.  The 
level  constant  must  be  found  as  just  described. 

4.  Latitude  by  Opposite  and  nearly  equal  Meridian  Zenith  Dis- 
tances. Talcott's  Method.  See  Form  9. 

This  method  depends  upon  the  principle  that  the  astronomical 
latitude  of  a  place  is  equal  to  the  declination  of  the  zenith. 

Let  zn  and  zg  represent  the  observed  meridian  zenith  distances 
of  two  stars,  the  first  north'  and  the  second  south  of  the  zenith  ; 
rn  and  rs  the  corresponding  refractions;  and  dn  and  ds  their  ap- 
parent decimations.  Then,  0  denoting  the  latitude, 

0  =*.  +  *.  +  r. ,  (141) 

</>=dn-zn-  rn.  (142) 

*     From  which 


Since  refraction  is  a  direct  function  of  the  zenith  distance,  this 
equation  shows  that  any  constant  error  in  the  adopted  refraction 
will  be  nearly  or  wholly  eliminated  if  we  select  two  stars  which 
culminate  at  very  nearly  the  same  zenith  distance,  and  provided 
also  that  the  time  between  their  meridian  transits  is  so  short  that 
the  refractive  power  of  the  atmosphere  cannot  be  changed  appre- 
ciably in  the  mean  time. 

Again,  since  absolute  zenith  distances  are  not  required,  but  only 
their  difference,  if  the  stars  are  so  nearly  equal  in  altitude  that  a 
telescope  directed  at  one,  will,  upon  being  turned  around  a  vertical 
axis  180°  in  azimuth,  present  the  other  in  its  field  of  view,  then 
manifestly  the  difference  of  their  zenith  distances  may  be  measured 
directly  by  the  declination  micrometer,  and  the  use  of  a  graduated 
circle  (with  its  errors  of  graduation,  eccentricity,  etc.)  be  entirely 


THE  ZENITH  TELESCOPE.  07 

dispensed  with,  except  for  the  purpose  of  a  rough  finder.  The  in- 
strument used  in  this  connection  is  called  a  "  Zenith  Telescope" 
Its  construction,  and  application  to  the  end  in  view,  are  best  learned 
from  an  examination  of  the  instrument  itself. 

Again,  since  errors  in  the  declinations  will  affect  the  resulting 
latitude  directly,  we  should  be  very  careful  to  employ  only  the  ap- 
parent declinations  for  the  date. 

The  following  conditions  should  therefore  be  fulfilled  in  select- 
ing the  stars  of  a  pair : 

1st.  They  should  culminate  not  more  than  20°,  or  at  most  25°, 
from  the  zenith. 

2d.  They  should  not  differ  in  zenith  distance  by  more  than 
15',  and  for  very  accurate  work,  by  not  more  than  10'.  The  field 
of  view  of  the  telescope  is  about  30'.  The  limit  assigned  prevents 
observations  too  near  the  edge  of  the  field,  and  lessens  the  effect  of 
an  error  in  the  adopted  value  of  a  turn  of  the  micrometer  head. 
This  limit  also  requires  a  very  approximate  knowledge  of  the  lati- 
tude, which  may  be  found  with  the  sextant,  or  by  measuring  the 
meridian  zenith  distance  of  a  star  by  the  zenith  telescope  itself. 

3d.  They  should  differ  in  R.  A.  by  not  less  than  one  minute 
of  time,  to  allow  for  reading  the  level  and  micrometer,  and  by  not 
more  than  fifteen  or  twenty  minutes,  to  avoid  changes  in  either  the 
instrument  or  the  atmosphere. 

Since  the  Ephemeris  stars,  whose  apparent  declinations  are 
given  with  great  accuracy  for  every  ten  days,  are  comparatively  few 
in  number,  it  becomes  necessary,  in  order  to  fulfil  the  above  con- 
ditions, to  resort  to  the  more  extended  star  catalogues. 

But  since  in  these  works  only  the  stars'  mean  places  are  given, 
and  those  for  the  epoch  of  the  catalogue  (which  fact  involves  re- 
duction to  apparent  places  for  the  date),  and  moreover  since  these 
mean  places  have  often  been  inexactly  determined,  it  becomes  de- 
sirable to  rest  our  determination  of  latitude  on  the  observation  of 
more  than  one  pair.  For  example,  on  the  "  Wheeler  Survey,"  west 
of  the  100th  meridian,  the  latitude  of  a  primary  station  was  re- 
quired to  be  determined  by  not  less  than  35  separate  and  distinct 
pairs  of  stars,  these  observations  being  distributed  over  five  nights. 

Preliminary  Computations. — We  should  therefore  form  a  list  of 
all  stars  not  less  than  7th  magnitude  which  culminate  not  more 
than  25°  from  the  zenith  and  within  the  limits  of  time  over  which 


98  PRACTICAL  ASTRONOMY. 

we  propose  to  extend  our  observations,  arrange  them  in  the  ordei 
of  their  R.  A.,  and  from  this  list  select  our  pairs  in  accordance 
with  the  above  conditions,  taking  care  that  the  time  between  the 
pairs  is  sufficient  to  permit  the  reading  of  the  level  and  micrometer, 
and  setting  the  instrument  for  the  next  pair;  say  at  least  two 
minutes. 

A  "Programme"  must  then  be  prepared  for  use  at  the  instru- 
ment, containing  the  stars  arranged  in  pairs,  with  the  designation 
and  magnitude  of  each  for  recognition  when  more  than  one  star  is 
in  the  field;  their  R.  A.,  to  know  when  to  make  ready  for  the 
observation;  their  declinations,  from  which  are  computed  their 
approximate  zenith  distances;  a  statement  whether  the  star  is  to  be 
found  north  or  south  of  the  zenith,  and  finally  the  "  setting  "  of  the 
instrument  for  the  pair,  which  is  always  the  mean  of  the  two 
zenith  distances. 

The  declinations  here  used,  being  simply  for  the  purpose  of  so 
pointing  the  instrument  that  the  star  shall  appear  in  the  field, 
may  be  mean  decimations  for  the  beginning  of  the  year,  which  are 
found  with  facility  as  hereafter  indicated.  Similarly  for  the  R.  A. 
For  this  Programme,  see  Form  9. 

Adjustment  of  Instrument. — The  Instrument  must  next  be  pre- 
pared for  use.  The  column  is  made  vertical  by  the  levelling 
screws,  and  the  adjustment  tested  by  noting  whether  the  striding 
level  placed  on  the  horizontal  axis  will  preserve  its  reading  during 
a  revolution  of  the  instrument  3GO°  in  azimuth.  The  horizontality 
of  the  latter  axis  is  secured  by  its  own  adjusting  screws,  and  tested 
by  the  level  in  the  usual  way.  The  focus  and  vertically  'of  the 
wires  are  adjusted  as  explained  for  the  transit.  The  collimation 
error  should,  as  far  as  is  mechanically  possible,  be  reduced  to  zero. 
This  may  be  accomplished  approximately  by  the  ordinary  reversals 
upon  a  terrestrial  point  distant  not  less  than  5  or  6  miles  (to 
reduce  the  parallax  caused  by  the  distance  of  the  telescope  from 
the  vertical  column);  or  very  perfectly  by  two  collimating  tele- 
scopes, as  explained  for  the  transit.  The  instrument  is  adjusted  to 
the  meridian  as  explained  for  the  transit.  When  this  is  perfected, 
one  of  the  movable  stops  on  the  horizontal  circle  is  moved  up  against 
one  side  of  the  clamp  which  controls  the  motion  in  azimuth,  and  there 
fixed  by  its  own  clamp-screw.  The  telescope  is  then  turned  180° 
around  the  vertical  column  and  again  adjusted  to  the  meridian  by 


THE  ZENITH  TELESCOPE.  99 

a  circum-polar  star;  the  other  stop  is  then  placed  against  the  other 
side  of  the  clamp,  and  fixed.  The  instrument  can  now  be  turned 
exactly  180°  in  azimuth,  bringing  up  against  the  stops  when  in  the 
meridian. 

Observations. — The  circle  being  set  to  the  mean  of  the  zenith 
distances  of  the  two  stars  of  a  pair,  the  bubble  of  the  attached  level 
is  brought  as  nearly  as  possible  to  the  middle  of  its  tube,  and  when 
the  first  star  of  the  pair  arrives  on  the  middle  transit  wire  (the  in- 
strument being  in  the  meridian)  it  is  bisected  by  the  declination  mi- 
crometer wire,  the  sidereal  time  noted,  and  the  micrometer  and  level 
read.  The  telescope  is  then  turned  180°  in  azimuth,  the  clamp 
bringing  up  against  its  stop.  The  same  observations  and  records 
are  now  made  for  the  second  star.  The  instrument  is  then  reset  for 
the  next  pair,  and  so  on.  The  time  record  is  not  necessary  unless 
it  be  found  that  the  instrument  has  departed  from  the  meridian,  or 
unless  observation  on  the  middle  wire  has  been  prevented  by 
clouds,  and  it  becomes  desirable  to  observe  on  a  side  wire  rather 
than  lose  the  star.  In  these  cases  the  hour  angle  is  necessary  to* 
obtain  the  "reduction  to  the  meridian."' 

The  observations  are  recorded  on  Form  9  a.  In  the  column  of 
remarks  should  be  noted  any  failure  to  observe  on  middle  wire, 
weather,  and  any  circumstance  which  might  affect  the  reliability  of 
the  observations. 

Reduction  of  Observations. — By  referring  to  Eq.  (143)  the  gen- 
eral nature  of  the  reduction  will  be  evident.  The  principal  term 
in  the  value  of  0  is  dn  -\-  ds ,  which,  as  before  stated,  must  be  found 
for  the  date.  Since  zs  —  zn  has  been  measured  entirely  by  the  mi- 
crometer and  level,  this  term  involves  two  corrections  to  3n  -f-  $8 ; 
rs  —  f"n  involves  another,  and  the  very  exceptional  case  of  observa- 
tion on  a  side  wire  involves  another. 

1st.  The  reduction  from  mean  declination  of  the  epoch  of  the 
catalogue  to  apparent  declination  of  the  date.  Let  us  take  the 
case  of  the  B.  A.  C.  (British  Association  Catalogue). 

The  star's  mean  place  is  first  brought  up  to  the  beginning  of 
the  current  year  by  the  formula 


100  PRACTICAL  ASTRONOMY. 

In  which  d"  =  mean  north  polar  distance  as  given  in  catalogue, 
pr  =  annual  precession  in  N.  P.  distance,  s'  =  secular  variation  in 
same,  /*'  =  annual  proper  motion  in  N.  P.  distance  (all  given  in 
catalogue  for  each  star),  y  =  number  of  years  from  epoch  of  cata- 
logue to  beginning  of  current  year,  and  d"'  =  the  mean  N.  P.  dis- 
tance at  the  latter  instant.-  To  this,  jkhe  corrections  for  precession, 
proper  motion,  nutation,  and  aberration,  since  the  beginning  of  the 
year,  are  applied  by  the  formula 

d  =  d'"  +  r^'  +  Ac'  +  Bd'  +  Ca'  -  LV, 

in  which  t  =  fractional  part  of  year  already  elapsed  at  date,  given 
on  pp.  285-292,  Ephemeris;  A,  B,  C,  D,  are  the  Besselian  Star 
Numbers,  given  on  pp.  281-284  Ephemeris  for  each  day;  «',  V  ,  c'  , 
d',  are  star  constants,  whose  logarithms  are  given  in  the  catalogue; 
and  d  =  star's  apparent  N.  P.  distance  at  date.  Then  6  =  90°  —d. 

The  quantities  a',  V,  c',  d',  are  not  strictly  constant;  indeed 
many  of  their  values  have  changed  perceptibly  since  1850,  the 
epoch  of  B.  A.  C.  If  it  be  desired  to  obviate  this  slight  error,  it 
may  be  done  by  recomputing  them  by  formulas  derived  from 
Physical  Astronomy,  or,  in  part,  by  using  a  later  catalogue.  In 
this  connection  a  work  prepared  under  the  "  Wheeler  Survey," 
entitled  "  Catalogue  of  Mean  Declinations  of  2018  Stars,  Jan.  1, 
1875,"  will  be  found  most  convenient,  embracing  stars  between  10° 
and  70°  N.  Dec.,  and  therefore  applicable  to  the  whole  area  of  the 
IT.  S.  exclusive  of  Alaska. 

With  this  catalogue,  the  reductions  are  made  directly  in  decli- 
nation, not  N.  P.  distance,  and  by  the  formulas, 


d=d'  +  Tn'  +  Aa'  +  BV  +  Cc'  +  Dd', 

in  which  everything  relates  to  declination. 

Exactly  analogous  formulas  hold  for  reduction  in  R.  A. 

J\  I  Ok 

2d.  The  micrometer  and  level  corrections  to  -      —  ?,  viz. 


THE  ZENITH  TELESCOPfi.  101 

Let  us  suppose  that,  with  the  telescope  set  at  a  given  inclina- 
tion, the  micrometer  readings  are  greater  as  the  body  viewed  is 
nearer  the  zenith;  and  in  the  first  instance,  that  the  inclination  as 
shown  by  the  attached  level  is  not  changed  when  the  instrument  is 
turned  180°  in  azimuth. 

Then  -?— — -  will  be  given  wholly  by  the  micrometer,  and  be 

.  ,      m,  —  mn           mn  —  ms  n 
either — "  R,  or  -  — R,  m  which  ms  and  mn  are  the  mi- 

*  /i 

crometer  readings  on  the  south  and  north  stars  respectively,  and  R 
the  value  in  arc  of  a  division  of  the  micrometer  head.  Since  the 
readings  increase  as  the  zenith  distance  decreases,  it  is  manifest 

that  ~^~ —  s  R  is  the  one  of  the  two  expressions  which  will  repre- 
sent —  with  its  proper  sign. 

But  as  a  rule  the  upright  column  will  not  be  truly  vertical,  and. 
therefore  the  inclination  of  the  optical  axis  of  the  telescope  will 
change  slightly  due  to  the  necessary  revolution  between  the  obser- 
vations of  the  stars  of  a  pair,— the  fact  being  indicated  by  a  different 
reading  of  the  level.  In  this  case,  the  difference  of  micrometer  read- 
ings will  not  be  strictly  the  difference  of  zenith  distance  as  before, 
but  will  be  that  difference  ±  the  amount  the  telescope  has  moved. 
The  micrometer  readings  therefore  require  correction  before  they 

can  give  -*—— —•  Since  it  is  immaterial  which  star  of  the  pair  is 
observed  first,  let  us  suppose  it  to  be  the  southern,  and  let  ln  and  la 
be  the  readings  of  the  ends  of  the  bubble.  Then  JL- — ?  will  be  the 

6 

reading  of  the  level,  it  being  graduated  from  the  center  toward 
each  end.  Now  if,  on  turning  to  the  north,  the  level  shows  that 
the  angle  of  elevation  of  the  telescope  has  increased,  the  microme- 
ter reading  on  the  northern  star  will  be  too  small,  by  just  the 
amount  corresponding  to  the  motion  of  the  telescope  in  altitude; 
arid  this  whether  the  star  be  higher  or  lower  than  the  southern 
star.  Consequently  mn  must  be  increased  to  compensate.  If  l'n 
and  1'8  be  the  reading  of  the  present  north  and  south  ends  of  the 


102  PRACTICAL  ASTRONOMY. 

V   —V 
bubble,  then  the  bubble  reading  will  be  -—  «  —  -8  j   the  change  of 

level,  in  level  divisions,  will  be 

±z±  +  ,  and  in  arc  (l  +  PJ-ft  +  rj  D 


Since,  upon  turning  to  the  north,  the  angle  of  elevation  of  the.  tele- 
scope was  supposed  to  increase,  this  quantity  is  positive;  and  being 
the  angular  change  of  elevation,  it  is  the  correction  to  be  applied 
to  mn. 

If  the  telescope  diminished  its  elevation  on  being  turned  to  the 
north,  it  would  be  necessary  to  diminish  mn  by  the  same  amount. 
But  in  this  case  the  above  correction  is  obviously  negative,  and  the 
result  will  be  obtained  by  still  adding  it  algebraically. 

The  correction  to---  will  be  half  the  above  amount;  hence  in 
ft 

all  cases  we  have  the  rule.  Subtract  the  sum  of  the  south  readings 
from  the  sum  of  the  north.  One-fourth  the  difference  multiplied 
by  the  value  of  one  division  of  the  level,  will  be  the  level  correction. 
The  true  difference  of  observed  zenith  distances  of  the  two  stars, 
is  therefore 


a    , 


T    •—  T 

3d.  The  correction  for  refraction,  or  -?L—  —  -.     Since  the  stars 

/o 

are  at  so  small  and  so  nearly  equal  zenith  distances,  differences  of 
actual  refractions  will  be  practically  equal  to  differences  of  mean 
refractions  (Bar.  30  in.,  F.  50°),  which  latter  may  therefore  be  substi- 

dr 

tuted  for  r8  —  rn  .    If  -r  denote  the  change  in  mean  refraction  for  a 

difference  of  1'  in  zenith  distance,  then  for  zs  —  zn  (expressed  in 
seconds)  it  will  be  s  n  -j  .  Hence  we  may  write 

rs  —  rn  _     z8  —  zn  dr 


60       dz' 


THE  ZENITH  TELESCOPE. 


103 


To  determine  -,-,  we  have  for   the   equation  of   mean   refraction 
Young,  p.  64), 

r  =  a  tan  z. 
Differentiating, 

dr  .         ,        a  sin  1' 
-T-  (for  1')  =  -  ~r-, 


a  being  taken  from  refraction  tables,  and  z  representing  the  mean 
of  the  zenith  distances  of  the  pair.     The  following  table  of  values 

dr 
:)f  -y-  is  given,  in  which  we  may  interpolate  at  pleasure. 


z 

dr 

dz 

0° 

0.0168" 

5° 

0.0169" 

10° 

0.0173" 

15° 

0.0180" 

20° 

0.0190" 

25° 

0.0205" 

The  principal  term  in  -^ — -  is  -n        -•  R.     Hence  we  may  write 

rs  —  rn  _  t  nin  —  ms     dr 
2         ~*       60        K~dz' 

and  the  correction  for  refraction  will  have  the  same  sign  as  the 
micrometer  correction. 

Hence  the  rule :  Multiply  the  micrometer  correction  in  minutes 

by  the  tabular  value  of  -j- ,  and  add  the  result  algebraically  to  the 

other  corrections. 

4th.  The  correction  to  the  zenith  distance  when  the  observation 
has  not  been  made  in  the  meridian;  i.e.,  when  not  made  on  the 
middle  vertical  wire. 

This  will  be  an  exceptional  correction,  but  one  which  must  oc- 
casionally be  made. 


104  PRACTICAL  ASTRONOMY. 

If  a  plane  be  passed  through  the  middle  horizontal  wire  and  the 
optical  centre  of  the  objective,  it  will  cut  from  the  celestial  sphere 
a  great  circle;  and  the  zenith  distance  of  a  star  anywhere  on  this 
circle  will,  as  measured  by  this  fixed  position  of  the  instrument,  be 
the  inclination  of  the  plane  to  the  vertical. 

Therefore,  if  the  zenith  distance'  of  a  star  between  the  zenith 
and  equinoctial  be  measured  by  an  instrument  which  moves  only 
in  the  meridian,  it  tvill  have  its  greatest  value  when  on  the  me- 
ridian. For  a  star  which  crosses  any  other  part  of  the  meridian, 
the  ordinary  rule  as  to  relative  magnitude  applies. 

But  whatever  the  position  of  the  star,  the  numerical  value  of 
this  "  reduction  to  the  meridian,"  due  to  an  observation  on  a  side 
wire,  is  different  from  that  heretofore  discussed,  where  the  instru- 
ment was  in  the  vertical  plane  of  the  star;  being  in  this  case 
£  (15  P)*  sin  1"  sin  2  tf;  P  being  the  hour  angle.  For  the  deduc- 
tion of  this  expression,  see  »J«  following.  For  a  star  below  the  equi- 
noctial or  below  the  pole  sin  2  $  would  be  negative;  hence  from  the 
rule  as  to  relative  magnitudes  above  given,  it  is  seen  that  if  in  using 
the  zenith  telescope,  a  star  south  of  the  zenith  be  observed  on  a  side 
wire,  the  above  correction  must  be  added  algebraically  to  the  ob- 
served to  obtain  the  meridian  zenith  distance;  and  north  of  the 
zenith  it  must  be  subtracted  algebraically. 

By  inspecting  the  term  ~^—^  —  -  ,  we  see  that  in  any  case  one  half 
this  reduction,  or 

i  (15  P)*  sin  1"  sin  2  d  =  [6.1347]  P2  sin  2  #, 

is  to  be  added  to  the  deduced  latitude,  or  to  the  sum  of  the  other 
corrections  in  order  to  obtain  the  latitude.  The  hour  angle  P  in 
seconds  of  time  is  known  from  P  =  t  -{-  E  —  oc,  t  being  the  chro- 
nometer time  of  observation,  E  the  error,  and  a  the  star's  R.  A. 
We  therefore  have  the  following  complete  formula  for  the  latitude 


.-.  ,         »-  . 

3-  -J- 

(144) 
~m*  Rd£+  [6.1347]  P'  sin  2  tfs  +  [6.1347]  P'2  sin  2  #„ 


THE  ZENITH  TELESCOPE.  105 

For  the  reduction  see  Form  9 b.     The  results  of  all  the  pairs  may 
be  discussed  by  Least  Squares. 

This  method,  although  extremely  simple  in  theory,  involves 
considerable  labor.  It  has  however  been  employed  almost  exclus- 
ively on  the  Coast  and  other  important  Government  surveys,  with 
results  which  compare  favorably  with  those  obtained  by  the  first- 
class  instruments  of  a  fixed  observatory. 


»J«  To  Determine  the  Reduction  to  the  Meridian  for  an  Instru- 
ment in  the  Meridian.  —  Let  S  Fig.  21  be  the  place  of  the  star  when 
on  a  side  wire.  Then  CSS"  will  be 
the  projection  of  the  great  circle  cut 
from  the  celestial  sphere  by  the  plane 
of  the  middle  horizontal  wire  and  the 
optical  center  of  the  objective,  Z  S" 
will  be  the  recorded  zenith  distance 


c 

diurnal  path,  preserving  always  the  FlG  21 

same    distance    from    the    equator. 

Then   Z  8'  will  be  the  true   meridian  zenith  distance  =  zt ,  and 

ES'  =  <?.     Represent  E  8"  by  d'. 

The  Reduction  to  the  Meridian,  /S"  8"9  being  denoted  by  x9  we 
have 

zt  =  z'  4-  x9    and     d  =  #'  —  x.  (a) 

Let  it  now  be  required  to  develop  x  into  a  series  arranged  according 
to  the  ascending  powers  of  sin2  J  P,  as  before. 
The  triangle  P  8  8"  9  right  angled  at  S"9  gives 

tan  6  =  cos  P  tan  d'  =  tan  <?'  —  2  tan  d'  sin8  £  P.  (5) 

Replacing  for  brevity  sin2 1  P  by  y, 

tan  d  =  tan  d'  —  2  ?/  tan  d',  (c) 


1  +  tan  d'  tan  x 
=  tan  d'  —  2  y  tan  d', 
tan  $'  —  tan  x  =  tan  6'  —  2  y  tan  d'  -\-  tan2  d'  tan  x 

-  2y  tan3  d'  tana;.  (d) 


106  PRACTICAL  ASTRONOMY. 

Let 

x  =  Ay  -\-  B  y*  -\-  etc.  (e) 

be  the  undetermined  development  desired.     If  the  value  of  tan  x 
derived  from  this  equation  be  substituted  in   (d),  the   resulting 
equation  will  be  identical. ' 
From  Trigonometry, 

tan  x  =  x-\--:  +  etc., 
o 

and  from  this  and  (e), 

tan  x  =  A  y  +  B  y*  +  etc. 

Substituting  in  (d),  and  transposing, 

Ay+By*-2  y  tan  tf'-ftan2  d'  (J«/+%a)-2tana  S'(Ay+By*)y=Q. 
From  the  principles  of  identical  equations, 

A  —  2  tan  c5>'  -}-  A  tan2  d'  =  0. 

2  tan  #'  sin  tf'  ,       . 

-4  =  .,—     — r^r,  =  2 -^  cos2  d'  =  sin  2  6^'. 

1  -f-  tan  o  cos  o 

B  +  B  tan2  d'  -  2  ^  tan2  tf'  =  0.         B  =  2  sin2  <S'  sin  2  d'. 
Therefore,  expressing  a;  in  seconds  of  arc,  from  (e), 
n  2  d'       2  sin4  i  P  sin  2  d'  sin2  6"' 


sin  1"  sin  1" 

Omitting  the  last  term  as  insensible,  expressing  P  in  seconds  of 
time,  and  remembering  that  since  P  is  very  small, 


sin3  =          -     sin2 1">  we  have  a;  =  i  (15  P)2  sin  1"  sin  2  d'. 

/O  \     4       J 

In  computing  this  term,  #  may  be  substituted  for  6^. 


t%t  To  Determine  the  Probable  Error  of  the  Final  Result. 
From  equation  (143)  it  is  seen  that  the  probable  error  of  a  lati- 
tude deduced  from  a  single  pair  of  stars  will  be  composed  of  two 


THE  ZENITH  TELESCOPE.  107 

parts:  1st,  the  probable  error  of  the  half  sum  of  the  declinations 
derived  from  the  catalogue  used;  2d,  the  probable  error  of  the  half 
difference  of  the  measured  zenith  distances,  which  may  be  called 
the  error  of  observation. 

Consider  first  a  single  pair  of  stars  observed  once.  Let  R^  de- 
note the  probable  error  of  the  deduced  latitude,  11'  that  of  the  half 
sum  of  the  declinations,  and  R"  that  of  observation,  all  unknown 
as  yet.  Then,  Johnson,*  Art.  89, 


-\-R"*,  (a) 

and  for  this  pair  observed  n  times,  i.e.)  on  n  nights, 


n 
If  now  we  employ  m  different  pairs, 


7?" 2 

(b) 


•¥  +  ~HT'  (c) 

in  which  n'  denotes,  as  before,  the  total  number  of  observations. 

It  may  be  observed  fit  this  point,  that  as  shown  by  (c),  if  a  skilled 
observer  be  provided  with  a  catalogue  not  of  the  first  order  of  ex- 
cellence, (R'  large,  R"  small),  it  is  better  to  employ  many  pairs, 
rather  than  repeat  observations  on  a  few  pairs;  thus  augmenting 
both  m  and  n,  instead  of  n  alone. 

To  determine  R",  form  the  differences  between  the  mean  of  all 
the  latitudes  resulting  from  the  first  pair  and  the  separate  latitudes 
from  that  pair. 

The  residuals  denoted  by  v'9  v",  v"',  etc.,  will  manifestly  be 
free  from  any  effect  of  error  in  the  half  sum  of  the  declinations 
employed.  Do  the  same  with  the  results  from  each  of  the  other 
pairs,  giving  v/,  v/' •  va',  v3" etc., 

Then,  Johnson,  Art.  138, 


=  ±  0.6745  |--.  (d) 

n  —  m  ^  ' 


*  Johnson's  "  Theory  of  Errors  and  Method  of  Least  Squares,"  1890. 


108  PRACTICAL  ASTRONOMY. 

The  value  of  R"  should  not  exceed  about  0".8,  and  cannot  be 
expected  to  fall  below  0".3.  On  the  Coast  Survey,  its  value  has 
usually  been  slightly  less  than  0".5. 

To  determine  R',  we  have  from  (b) 


in  which  it  must  be  remembered  that  R4  is  the  probable  error  of 
the  latitude  as  deduced  from  a  single  pair  of  stars  observed  n  times. 

Select  several  (mf)  pairs,  which  are  observed  on  an  equal  number 
of  nights  in  order  that  the  results  from  each  pair  may  be  of  equal 
weight.  Then,  as  before,  form  the  differences  between  the  mean 
of  the  n  results  for  each  pair  and  the  mean  of  these  m'  means. 

Then  the  mean  value  of  Rt  will  be,  Johnson,  Art.  72, 

Rt  =  0.6745  1/ mf_fr  (/) 

Substituting  this  value  of  Rt  together  with  that  of  n  in  (e),  we 
have  R',  and  the  probable  error  of  the  final  result  is  given  by  (c), 
as  before  seen. 

If  R'  be  determined  from  a  great  number  of  stars  taken  from  a 
single  catalogue,  it  may  be  considered  as  constant  for  that  cata- 
logue. With  the  one  employed  on  the  Lake  Survey,  R'  usually  fell 
between  0".53  and  0".60. 

If  it  be  desired  to  combine  the  mean  results  from  each  pair  ac- 
cording to  their  weights  in  order  to  obtain  the  weighted  mean 
latitude,  we  have  from  (6),  (since  the  weight  of  an  observation  is 
proportional  to  the  reciprocal  of  the  square  of  the  probable  error,) 

n 


p  denoting  the  weight  of  the  mean  result  from  a  pair  observed  n 
times. 

The  weighted  mean  latitude  will  be,  Johnson,  Art.  66, 

212T 


THE  ZENITH  TELESCOPE.  109 

with  a  probable  error,  Johnson,  Art.  72, 

R  =  0.6745 


The  errors  which  give  rise  to  R'  are  those  pertaining  to  the 
catalogue  or  catalogues  used. 

Those  giving  rise  to  R"  are  due  to  various  causes,  viz. :  imper- 
fect bisection  of  one  or  both  stars  due  to  personal  bias  or  unsteadi- 
ness of  the  stars,  anomalous  refraction,  errors  in  determining  the 
value  of  a  division  of  the  micrometer  and  level,  changes  in  temper- 
ature affecting  the  instrument  between  the  two  observations  of  a 
pair,  etc. 

If  any  of  the  residuals  (v)  are  unusually  large,  they  should  be 
examined  by  Peirce's  Criterion  bofore  rejection. 

Finally  it  must  be  remembered  that  in  this,  as  in  all  other 
methods  here  given,  the  final  result  (supposed  free  from  error)  is 
the  astronomical  latitude,  and  will  differ  from  the  geodetic  or  geo- 
graphical latitude  by  any  abnormal  deflection  of  the  plumb-line 
which  may  exist  at  the  station. 


5.  Latitude  by  Polaris  off  the  Meridian.     See  Form  10.— This 

method  depends  upon  the  fact  that  the  astronomical  latitude  of  a 
place  is  equal  to  the  altitude  of  the  elevated  pole. 

This  latter  is  obtained  by  measuring  the  altitude  of  Polaris  at  a 
given  instant,  and  from  the  data  thus  obtained,  together  with  the 
star's  polar  distance,  passing  to  the  altitude  of  the  pole. 

To  explain  this  transformation : 

Let  P  =  star's  hour  angle,  measured  from  the  upper  meridian. 
a  =  altitude  of  star  at  instant  P,  corrected  for  refraction. 
d  =  polar  distance  of  star  at  instant  P. 
0  =  latitude  of  place. 
Then  from  the  Z  P  S  triangle  we  have 

sin  a  =  sin  0  cos  d  +  cos  0  sin  d  cos  P.  (145) 

This  equation  which  applies  to  any  star  may  be  solved  directly; 
but  with  a  circum-polar  star  it  is  much  simpler  to  take  advantage 
of  its  small  polar  distance,  and  obtain  a  development  of  0  in  terms 


110  PRACTICAL  ASTRONOMY. 

of  the  ascending  powers  of  d,  in  which  we  may  neglect  those  terms 
which  can  be  shown  to  be  unimportant. 

Now  if  we  let  x  —  the  difference  in  altitude  between  Polaris  at 
the  time  of  observation  and  the  pole,  we  shall  have 

0  =  (a  —  x),   sin  0  =  sin  (a  —  x),     cos  0  =  cos  (a  —  x), 
and  from  (145), 

1  =  cos  x  (cos  d  +  sin  d  cot  a  cos  P)  ,       , 

—  sin  x  (cos  d  cot  a  —  sin  d  cos  P). 

Moreover,  it  is  evident  that  if  we  can  obtain  the  development  of 
x  in  terms  of  the  ascending  powers  of  d,  we  will  have  the  develop- 
ment of  0  in  the  same  terms,  from  0  =  a  —  x. 

This  is  the  end  to  be  attained.     Therefore  let 

x  =  A  d  +  B  d*  +  Cd*  +  etc.,  (147) 

be  the  undetermined  development  desired,  in  which  A,  B,  C,  etc., 
are  to  have  such  constant  values,  that  the  series,  when  it  is  con- 
vergent, shall  give  the  true  value  of  x,  whatever  may  be  the  value 
of  d. 

It  is  manifest,  then,  that  if  this  assumed  value  of  x  be  substi- 
tuted in  (146),  the  resulting  equation  must  be  satisfied  by  every 
value  of  d  which  renders  (147)  convergent;  that  is,  the  resulting 
equation  must  be  identical;  otherwise  (147)  could  not  be  true. 

With  a  view,  therefore,  to  this  substitution,  let  it  be  noted  that 
by  the  Calculus  we  have 

x1       x* 
=  l--+  —  -  etc.,  (m) 


-  +         -  etc.,  (n) 


and  hence  from  (147), 


cos  x  =  i  -      -  -AB<F.+  etc.,  (us) 

a 


--    d3  +  etc.  (149) 


LATITUDE  BY  POLARIS. 


Ill 


Also, 

\ 

and 


cos  d  =  1  -  -2  +  24  -  etc., 


(150) 
(151) 


Now  d  is  a  very  small  angle;  at  present  about  1°  16',  or  0.0221 
radians;  x  can  never  be  greater  than  d,  and  in  the  general  case 
w  ill  be  less.  Under  these  circumstances  the  above  series  becomes 
very  convergent,  and  the  sum  of  a  few  terms  will  represent  with 
great  accuracy  the  sum  of  the  series.  It  is  -for  this  reason  that  the 
problem  under  discussion  is  applicable  only  to  close  circum-polar 
stars,  and  therefore  we  take  advantage  of  the  small  polar-distance 
of  Polaris. 

Substituting  (148),  (149),  (150),  and  (151),  in  (146),  we  have, 
rejecting  terms  involving  the  4th  and  higher  powers  of  d, 

A  cot  a 


+  £cosP 

cos  P  cot  a 
6 

A1  cos  P  cot  a 

~1T~ 

-AB 

_  (c-~\cota 


AcosP 
—  £  cot  a 


cos  P  cot  a 


—  A  cot  a 


d=0. 


This   equation  being  identical,  the  algebraic  sum  of  the  coeffi- 
cients of  each  power  of  d  must  be  separately  equal  to  zero. 
Hence  we  have  by  solution, 

A  =  cos  P. 

sin2  P  , 
B— ~  -  tan  0. 


C  = 


cos  P  sin2  P 


112  PRACTICAL  ASTRONOMY. 

Therefore,  from  (147), 

x  =  d  cos  P  —  %  d*  sin*  P  tan  a  +  J  d3  cos  P  sin2  P. 

From  (m)  and  (w),  (150)  and  (151),  it  is  seen  that  x  and  d  are 
expressed  in  radians.  Expressing  them  in  seconds  of  arc, 

x  =  d  cosP-i  d*  sin2  P  tan  a  sin  1"+  J  d3  cos  P  sin'P  sin2 1",  etc., 

which  is  the  required  development. 
Therefore, 

0  =  a  —  d  cos  P  -j-  J  rf3  sin  1"  sin2  P  tan  a 

-  J.  d3  sin2 1"  cos  P  sin'  P  |-  etc.  (152) 

The  last  three  terms  are  in  seconds. 

Hence  we  have  the  general  rule: 

Take  a  series  of  altitudes  of  Polaris  at  any  convenient  time. 
Note  the  corresponding  instants  by  a  chronometer,  preferably 
sidereal,  whose  error  is  well  determined.  Correct  each  observed 
altitude  for  instrumental  errors  and  refraction.  Determine  each 
hour  angle  by  P  =  sidereal  time  —  R.  A. 

Take  from  the  Ephemeris  the  star's  polar  distance  at  the  time, 
being  careful  to  use  pp.  302-313,  where  also  the  R.  A.  required 
above  will  be  found. 

Substitute  each  set  of  values  in  Equation  (152),  and  reduce  each 
set  separately.  The  mean  of  the  resulting  values  of  0  is  the  one 
adopted.  See  Form  10. 

As  before  stated,  the  method  is  applicable  only  to  close  circum- 
polar  stars.  Polaris  is  selected  since  it  is  the  nearest  bright  star  to 
the  pole,  a  fact  which  is  of  importance  in  sextant  observations. 

On  the.  last  page  of  the  Ephemeris  are  given  tabular  values  of 
the  correction  x.  They  are  however  only  approximate;  and  the 
complete  solution,  as  given  above,  consumes  but  very  little  more 
time. 

This  is  a  very  convenient  method  of  determining  latitude;  our 
only  restriction  being  that,  with  a  sextant,  the  observations  must 
be  made  at  night.  With  the  "  Altazimuth  "  instrument,  the  ob- 
servations may  be  made  for  some  time  before  dark. 


LATITUDE  BY  POLARIS.  113 

The  last  term  in  (152)  is  very  small.  In  order  to  ascertain 
whether  it  is  of  any  practical  value,  let  us  determine  its  maximum 
numerical  value.  Denoting  the  term  by  z,  and  its  constant  factors 
by  c,  we  have 

z  =  c  cos  P  sin2  P. 

Replacing  sin2  P  by  1  —  cos2  P,  and  differentiating  twice,  we  have, 
after  reduction, 

(1  Z 

-  3  c  sin3  P. 


z 

=  2  c  cos  P  —  9  c  sin2  P  cos  P. 


—  — 

To  obtain  the  maximum, 

2  c  sin  P  -  3  c  sin3  P  =  0. 
From  the  roots  of  this  we  have 

sin  P  =  0,     sin  P  =  +  \T%,     sin  P  =  —  4/~f  , 

the  last  two  of  which  correspond  to  equal  numerical  maxima.  * 
Hence  the  maximum  value  of  the  term  is  given  when  sin2  P  =  f, 
or  when  z  =  $  d3  sin2  I"  f  /|.  For  d  =  1°  16',  this  gives  z  =  0".29. 
The  maximum  error  committed  by  the  omission  of  this  term 
will  therefore  be  about  0".3.  Evidently  its  retention  when  the 
observations  have  been  made  with  a  sextant  would  be  superfluous. 

*  With  sin  P  =  -j-  |/  f  \ve  may  have  cos  P  =  ±  4/£  ,  and  similarly  for 
siuP  =  —  |/f  .  By  substituting  in  the  second  differential  coefficient  we  see 
that  ±  -f/f  with  -|-  V  %  correspond  to  equal  maxima,  while  ±  |/~|  with  —  \/~% 
correspond  to  equal  minima.  With  sin  P  =  0,  we  may  have  cos  P—  ±  1  ,  the 
former  of  which  corresponds  to  a  minimum  and  the  latter  to  an  equal  maxi- 
mum; viz.,  zero.  Hence  zero  is  a  lesser  and  not  the  greatest  maximum  value 
of  z  ;  the  latter,  with  which  only  we  are  concerned,  being,  from  (15:2),  i  d3  sin2 


Fig.  22  gives  the  curve  of  values  of  z  with  P  as  the  abscissae,  showing 
the  inferior  maximum  at  P  =180°,  and  the  greatest  maxima  (numerical)  at 
about  55°,  125°,  235°,  and  305°. 


114  PRACTICAL  ASTRONOMY. 

The  value  of  log  sin  1",  not  given  in  ordinary  tables,  is 
4.6855575-10. 

•v 

125°                     180°                     235° 
1 _  i..    1 ^f. 

./^      ^v  >1WO  305"  360" 

FIG.  22. 

Any  mistake  as  to  the  value  of  P  will  manifestly  produce  ite 
greatest  effect  when  the  star  is  moving  wholly  in  altitude.  Hence 
if  the  chronometer  error  be  not  well  determined,  the  times  of 
elongation  are  the  least  advantageous  for  observation. 

Since  cos  (360°  —  P)  =  cos  P,  we  may  measure  P  from  the  up- 
per meridian  to  180°  in  either  direction. 

6.  Latitude  by  Equal  Altitudes  of  Two  Stars.     See  Form  11. — 
By  this  method  the  latitude  is  found  from  the  declinations  and  hour 
angles  of  two  stars;  the  hour  angles  being  subject  to  the  condition 
that  they  shall  correspond  to  equal  altitudes  of  the  stars. 
Let  6  and  6'  =  the  correct  sidereal  times  of  the  observations. 

a  and  a'  =  the  apparent  right  ascensions  of  the  stars. 

6  and  d'  =  the  apparent  declinations  of  the  stars. 

P  and  P'  =  the  apparent  hour  angles  of  the  stars. 

a  =  the  common  altitude. 

0  =  the  required  latitude. 

P  and  P'  are  given  from 

P=0-a.  P'  =  6'  -a'. 

From  the  Z  P  S  triangle  we  have 

sin  a  =  sin  0  sin  d  -f  cos  0  cos  tf  cos  P. 
sin  a  =  sin  0  sin  6'  -j-  cos  0  cos  $'  cos  P'. 

Subtracting  the  first  from  the  second  and  dividing  by  cos  0, 
tan  0  (sin  d'  —  sin  6)  =  cos  d  cos  P  —  cos  6'  cos  P'.     (153) 

The  value  of  tan  0  might  be  derived  at  once  from  this  equa- 
tion, since  it  is  the  only  unknown  quantity  entering  it.  The  form 


LATITUDE  ST  EQUAL  ALTITUDES.  115 

is,  however,  unsuited  to  logarithmic  computation.  In  order  to  ob- 
tain a  more  convenient  form,  observe  that  the  second  member  may 
be  written 

/  cos  6  cos  P       cos  d'  cos  P'\       /cos  d  cos  P      cos  d'  cos  P' 
I  o  o  I    I    V  o  f> 

Adtiing  to  the  first  parenthesis 

/  cos  d  cos  P'        cos  £'  cos 


V          2  2 

and  subtracting  the  same  from  the  second,  we  have,  after  factoring, 

tan  0  (sin  tf '  —  sin  6)  =  J  (cos  d  —  cos  6')  (cos  P  -f  cos  P') 
+  |  (cos  d  +  cos  £')  (cos  P  —  cos  P'). 

Solving  with  reference  to  tan  0,  and  reducing  by  Formulas  16,  17, 
and  18,  Page  4,  Book  of  Formulas, 

tan  0  =  tan  \  (6'  +  3)  cos  J  (P'  +  P)  cos  J  (P'  -  P) 
+  cot  J  (d'  -  d)  sin  i  (P'  +  P)  sin  £  (P'  -  P). 

The  solution  may  be  made  even  more  simple  by  the  use  of  two 
auxiliary  quantities,  m  and  M,  such  that 

m  cos  M=  cos  J  (P'  —  P)  tan  |  (<?'  -f  6).  (155) 

m  sin  ^f  =  sin  J-  (P'  —  P)  cot  J  (<T  —  d)  (156) 

Then 

tan  0  =  m  cos  [|  (P'  +  P)  -  J/].  (157) 

Equations  (155)  and  (156)  give  m  and  J/,  and  (157)  gives  0,  all 
in  the  simplest  manner. 

For  example,  to  find  M,  divide  (156)  by  (155),  and  we  obtain 

tan  M  =  tan  \  (P'  —  P)  cot  J  (£'  —  d)  cot  J-  (£'  -J-  6). 

This  admits  of  easy  logarithmic  solution. 

The  value  of  m  follows  from  either  (155)  or  (156),  and  that  of 
0  from  (157),  both  by  logarithms. 


116  PRACTICAL  ASTRONOMY. 

The  value  of  a  does  not  enter;  hence  the  resulting  latitude  will 
be  entirely  free  from  instrumental  errors,  those  of  graduation,  ec- 
centricity, and  index  error,  and  its  accuracy  will  depend  only  upon 
the  skill  of  the  observer,  and  the  accuracy  of  our  assumed  chronom. 
eter  error  and  rate.  Ephemeris  stars  should  be  chosen  if  possible,  for 
the  sake  of  accuracy  in  declinatipnSj^and  their  E.  A.  should  permit 
the  observations  to  be  made  with  so 'short  an  interval  that  the  re- 
fractive power  of  the  atmosphere  can  not  have  changed  materially 
in  the  mean  time.  The  value  of  refraction  is  not  required ;  it  is 
only  necessary  that  it  remain  practically  constant. 

Differentiating  (153)  with  reference  to  0 ,  P,  and  P'9  solving, 
reducing  by 

cos  6'  sin  Pf  —  cos  a  sin  A', 
and 

sin  6'  =  sin  0  sin  a  -f-  cos  0  cos  a  cos  A1 ', 

we  have,  since  a  is  the  same  for  both  stars, 

7  ^  sin  A'          ?  r>/  sin  ^4 

d  0  =  cos  0  -   — -j—        —  d  P'  —  cos  0  —    —  —       —  d  P9 
cos^i  —  cos^4  cos  ^4   —  cos  A 

from  which  it  is  seen  that  any  error  in  the  time  or  in  the  assumed 
chronometer  correction  will  have  least  effect  on  the  resulting  latitude 
when  the  two  stars  reach  the  common  altitude  at  about  equal  dis- 
tances north  and  south  of  the  prime-vertical,  the  nearer  to  the 
meridian  the  better. 

When  several  observations  with  the  sextant  are  taken  in  succes- 
sion on  each  star,  it  is  better  to  reduce  separately  the  pair  corre- 
sponding to  each  altitude. 

LONGITUDE. 

The  difference  of  Astronomical  Longitude  between  two  places  is 
the  spherical  angle  at  the  celestial  pole  included  between  their  re- 
spective meridians.  By  the  principles  of  Spherical  Geometry,  the 
measure  of  this  angle  is  the  arc  of  the  equinoctial  intercepted  by  its 
sides;  or  it  is  the  same  portion  of  360°  that  this  arc  is  of  the  whole 
great  circle. 

But  since  the  rotation  of  the  earth  upon  its  axis  is  perfectly 
uniform,  the  time  occupied  by  a  star  on  the  equinoctial  in  passing 


LONGITUDE.  117 

from  one  meridian  to  another,  is  the  same  portion  of  the  time  re- 
quired for  a  complete  circuit  that  the  angle  between  the  meridians 
is  of  360°,  or,  that  the  intercepted  arc  is  of  the  whole  great  circle. 
Moreover,  all  stars  whatever  their  position  occupy  equal  times  in 
passing  from  one  meridian  to  another  due  to  the  fact  that  all  points 
on  a  given  meridian  have  a  constant  angular  velocity. 

The  same  facts  apply  also  to  the  case  of  a  body  which,  like  the 
mean  sun,  has  a  proper  motion,  provided  that  motion  be  uniform 
and  in  the  plane  of,  or  parallel  to,  the  equinoctial. 

Hence  it  is  that  Longitude  is  usually  expressed  in  time;  and  in 
stating  the  difference  of  longitude  between  two  places  in  time,  it  is 
immaterial  whether  we  employ  sidereal  or  mean  solar  time :  for  the 
number  of  mean  solar  time  units  required  for  the  mean  sun  to  pass 
from  one  meridian  to  another,  is  exactly  equal  to  the  number  of 
sidereal  time  units  required  for  a  star  to  pass  between  the  meridians. 

The  astronomical  problem  of  longitude  consists,  therefore,  in 
determining  the  difference  of  local  times,  either  sidereal  or  mean 
solar,  which  exist  on  two  meridians  at  the  same  absolute  instant. 

Since  there  is  no  natural  origin  of  longitudes  or  circle  of  refer- 
ence as  there  is  in  case  of  latitude,  one  may  be  chosen  arbitrarily, 
and  which  is  then  called  the  "  first "  or  "  prime  meridian."  Differ- 
ent nations  have  made  different  selections:  but  the  one  most  com- 
monly used  throughout  the  world  is  the  upper  meridian  of  Green- 
wich, England,  although  in  the  United  States  frequent  reference  is 
made  to  the  meridian  of  Washington.  , 

The  astronomical  may  differ  slightly  from  the  geodetic  or  geo- 
graphical longitude,  for  reasons  given  under  the  head  of  latitude. 

In  the  following  pages,  only  the  former  is  referred  to;  it  is 
usually  found  from  the  difference  of  time  existing  on  the  two 
meridians  at  the  instant  of  occurrence  of  some  event,  either  celes- 
tial or  terrestrial.  Up  to  about  the  year  1500  A.D.,  the  only  method 
available  was  the  observation  of  Lunar  Eclipses.  But  with  the 
publication  of  Ephemerides  and  the  introduction  of  improved 
astronomical  instruments,  other  and  better  methods  have  superseded 
this  one,  of  which  the  two  most  accurate  and  most  generally  used 
are  the  "  Method  by  Portable  Chronometers,"  and  the  "  Method  by 
Electric  Telegraph."  Longitude  may  also  be  found  from  "  Lunar 
Culminations  "  and  "  Lunar  Distances,"  in  cases  when  other  modes 
are  not  available. 


118  PRACTICAL  ASTRONOMY. 

1.  By  Portable  Chronometers.  Let  A  and  B  denote  the  two  sta- 
tions the  difference  of  whose  longitude  is  required.  Let  the  chron- 
ometer error  (E)  be  accurately  determined  for  the  chronometer  time 
T,  at  one  of  the  stations,  say  A  ;  also  its  daily  rate  (r). 

Transport  the  chronometer  to  B,  and  let  its  error  (E')  on  local 
time  be  there  accurately  determined  for  the  chronometer  time  T'. 
Let  i  denote  the  interval  in  chronometer  days  between  Tand  T'. 

Then,  if  r  has  remained  constant  during  the  journey,  the  true 
local  time  at  A  corresponding  to  the  chronometer  time  T'  will  be, 
T'  +  E  +ir. 

The  true  time  at  B  at  the  same  instant  is,  T'  4-  E'. 

Their  difference  =  difference  of  Longitude  is 

\=E+ir-E'.  (158) 

Thus  the  difference  of  Longitude  is  expressed  as  the  difference 
between  the  simultaneous  errors  of  the  same  chronometer  upon  the 
local  times  of  the  two  meridians,  and  the  absolute  indications  of 
the  chronometer  do  not  enter  except  in  so  far  as  they  may  be  re- 
quired in  determining  i. 

The  rule  as  to  signs  of  E  and  r,  heretofore  given,  must  be  ob- 
served. If  the  result  be  positive,  the  second  station  is  west  of  the 
first ;  if  negative,  east. 

This  method  is  used  almost  exclusively  at  sea,  except  in  voyages 
of  several  weeks,  the  chronometer  error  on  Greenwich  time,  and  its 
rate,  being  well  determined  at  a  port  whose  longitude  is  known. 
Time  observations  are  then  made  with  a  sextant  whenever  desired 
during  the  voyage,  and  the  longitude  found  as  above.  The  same 
plan  may  evidently  be  followed  in  expeditions  on  land,  although  ex- 
treme accuracy  cannot  be  obtained  since  a  chronometer's  "  travel- 
ing rate  "  is  seldom  exactly  the  same  as  when  at  rest. 

In  the  above  discussion,  the  rate  was  found  only  at  the  initial 
station.  If  the  rate  be  determined  again  upon  reaching  the  final 
station,  and  be  found  to  have  changed  to  r',  then  it  will  be  better  to 

r  -f  r'  . 
employ  in  the  above  equation  — - —  instead  of  r.    To  redetermine 

^  :— 

the  longitude  of  any  intermediate  station  in  accordance  with  this 

r'  —  r 
additional  data,  we  have  x  =  — -. —  =  daily   change   in  rate;   and 


LONGITUDE.  119 

the  accumulated  error  at  any  station,  reached  n  days  after  leaving  A, 
would  be  E  -f  f  r  -f  x  ~  J  w,  the  quantity  in  parenthesis  being  the 

rate  at  the  middle  instant. 

The  above  method  is  slightly  inaccurate,  since  we  have  assumed 
that  the  chronometer  rate  as  determined  at  one  of  the  extreme 
stations  (or  both,  if  we  apply  the  correction  just  explained),  is  its 
rate  while  en  route.  This  is  not  as  a  rule  strictly  correct. 

Therefore,  when  the  difference  of  longitude  between  two  places 
is  required  to  be  found  with  great  precision,  "  Chronometric  Expe- 
ditions "  between  the  points  are  organized  and  conducted  in  such  a 
manner  as  to  determine  this  traveling  rate. 

As  before, 

let  E     =  chron.  error  on  local  time  at  A  at  chron.  time  T. 

((      fff       __         «  t(  «  a  »  ((  (t         rpt 

tf         tr    _         ((  ((  ((  se  t<  ((  ' 


ft      TjiFH  —-         ((  (( 


ft  A  <( 


That  is,  the  error  on  local  time  is  determined  at  the  first  station  for 
the  time  of  departure,  then  at  the  second  station  for  the  time  of  ar- 
rival; again  at  the  second  station  for  the  time  of  departure,  and 
finally  at  the  first  station  for  the  time  of  arrival. 

Then  the  entire  change  of  error  is  E'"  —  E.  But  of  this 
E"  —  E'  accumulated  while  the  chronometer  was  at  rest  at  the 
second  station.  The  entire  time  consumed  was  T'"—T.  But  of 
this  T"  —  T'  was  not  spent  in  traveling.  Therefore,  the  traveling 

rate,  if  it  be  assumed  to  be  constant,  will  be 

' 


_ 

- 


_   rp\   _  /  m//  _    rp  r/\- 


This,  then,  is  the  rate  to  be  employed  in  Eq.  (158)  instead  of  the 
stationary  rate  there  used. 

If  the  rate  has  not  been  constant,  but,  as  is  often  the  case,  uni- 
formly increasing  or  decreasing,  the  above  value  of  r  is  the  average 
rate  for  the  whole  traveling  time  of  the  two  trips,  whereas  for  use 
in  Eq.  (158),  we  require  the  average  rate  during  the  trip  from  A  to 
B.  This  latter  average  will  give  a  perfectly  correct  resiiifc  provided 
the  rate  change  uniformly.  If  the  rate  has  been  increasing,  then  r 


120  PRACTICAL  ASTRONOMY. 

in  Eq.  (159)  will  be  too  large  numerically,  by  some  quantity  as  x. 
Hence  Eq.  (158)  becomes 

K  =  E+i(r-x)-E',  (160) 

in  which  r  is  found  by  (159).  In  order  to  eliminate  x,  let  the 
chronometer  be  transported  from  B  to  A,  and  return;  i.e.,  take  B 
instead  of  A  as  the  initial  point  of  a  second  journey.  This  is  best 
accomplished  by  utilizing  the  return  trip  of  the  journey  A  B  A,  as 
the  first  trip  of  the  journey  B  A  B. 

Then  the  new  average  rate  r'  having  been  found  as  before,  it 
will,  if  the  trips  and  the  interval  of  rest  have  been  practically  equal 
to  those  of  the  first  journey,  exceed  the  value  required,  by  the  same 
quantity,  x,  due  to  the  uniformity,  in  the  rate's  change.  Hence  for 
this  journey  Eq.  (158)  becomes, 

A  =  E'"  -  [i  (r'  -  x)  +  J0"]-  (161) 

In  the  mean  of  (160)  and  (161),  x  disappears,  giving, 

x=^+^+i^l_^±_^       (162) 

Hence,  if  our  time  observations  are  accurate,  and  the  traveling 
rate  constant,  the  difference  of  longitude  between  A  and  B  may  be 
determined  by  transporting  the  chronometer  from  A  to  B,  and  re- 
turn. Or,  if  the  rate  be  uniformly  increasing  or  decreasing,  the 
difference  of  longitude  will  be  found  by  transporting  the  chro- 
nometer from  A  to  B,  and  return,  then  back  tc  B\  thus  making 
three  trips  for  the  complete  determination. 

In  a  complete  "  Chronometric  Expedition,"  however,  many 
chronometers,  sometimes  60  or  70,  are  used,  to  guard  against  acci- 
dental errors ;  and  they  are  transported  to  and  fro  many  times.  As 
an  example,  in  one  determination  of  the  longitude  of  Cambridge, 
Mass.,  with  reference  to  Greenwich,  44  chronometers  were  employed 
and  during  the  progress  of  the  whole  expedition,  more  than  400 
exchanges  of  chronometers  were  made. 

They  are  rated  by  comparison  with  the  standard  observatory 
clocks  at  each  station,  which  are  in  turn  regulated  by  very  elabo- 
rately reduced  observations  on,  as  near  as  possible,  the  same  stars. 


LONGITUDE.  121 

Conducted  as  above  described,  " Chronometric  Expeditions'5 
give  exceedingly  accurate  results,  especially  if  corrections  be  made 
for  changes  in  temperature  during  the  journeys. 

2.  Longitude  by  the  Electric  Telegraph.  See  Form  12. — This 
method  consists,  in.  outline,  in  comparing  the  times  which  exist 
simultaneously  on  two  meridians,  by  means  of  telegraphic  signals. 
These  signals  are  simply  momentary  "  breaks"  'in  the  electric  cir- 
cuit connecting  the  stations,  the  instants  of  sending  and  receiving 
which  are  registered  upon  a  chronograph  at  each  station.  Each 
chronograph  is  in  circuit  with  a  chronometer  which,  by  breaking 
the  circuit  at  regular  intervals,  gives  a  time  scale  upon  the  chrono- 
graph sheet,  from  which  the  instants  of  sending  and  receiving  are 
read  off  with  great  precision. 

Suppose  a  signal  to  be  made  at  the  eastern  station  (A)  at  the 
time  T  by  the  clock  at  A,  which  signal  is  registered  at  the  western 
station  (B)  at  the  time  T'  by  the  clock  at  B. 

Then  if  E  and  E '  are  the  respective  clock  errors,  each  on  its 
own  local  time;  and  if  the  signals  were  recorded  instantly  at  B9 
then  the  difference  of  longitude  would  be  (T  +  E)  —  (T'  -f  E '). 
But  it  has  been  found  in  practice  that  there  is  always  a  loss  of  time 
in  transmitting  electric  signals.  Therefore  in  the  above  expression 
(Tf  -\-  E')  does  not  correspond  to  the  instant  of  sending  the  signal, 
but  to  a  somewhat  later  instant.  It  is  therefore  too  large,  the  entire 
expression  is  too  small,  and  must  be  corrected  by  just  the  loss  of 
time  referred  to.  This  is  usually  termed  the  "  Retardation  of  Sig- 
nals;" and  if  it  be  denoted  by  x,  the  true  difference  of  longitude 
will  be  (T+E)-(T'  +  Ef)  +  x  =  X'+x  =  X.  But  x  is  un- 
known, and  must  therefore  be  eliminated. 

In  order  to  do  this,  let  a  signal  be  sent  from  the  western  station 
at  the  time  T"  which  is  recorded  at  the  eastern  at  the  time  T'". 
Then  if  En  and  E'"  are  the  new  clock  errors,  the  true  difference 
of  longitude  will  be 

(T9"  +  E'")  -  (T"  +  E"}  -x  =  X"-x  =  l. 

By  addition,  x  disappears,  and  if  A  denote  the  longitude,  we  will 
have 


122 


PRACTICAL  ASTRONOMY. 


Or,  in  full,  assuming  that  the  errors  do  not  change  in  the  interval 
between  signals, 


(E- 


(163) 


T9  T',  T",  and  T"9  are  given  by  the  chronograph  sheets; 
E  and  Ef  must  be  determined  with  extreme  accuracy,  since  incor- 
rect values  will  affect  the  resulting  longitude  directly. 

Having  established  telegraphic  communications  between  the 
two  observatories  (field  or  permanent),  usually  by  a  simple  loop  in 
an  existing  line,  preliminaries  as  to  number  of  signals,  time  of 
sending  them,  intervals,  calls,  precedence  in  sending,  etc.,  are 
settled.  At  about  nightfall  messages  are  exchanged  as  to  the  suita- 
bility of  the  night  for  observations  at  the  two  stations.  If  suitable 
at  both)  each  observer  makes  a  series  of  star  observations  with  the 
transit  to  find  his  chronometer  error.  The  electric  apparatus  for 
this  purpose,  consisting  of  two  or  three  galvanic  cells,  a  break-cir- 
cuit key,  chronograph,  and  break-circuit  chronometer,  is  arranged 
as  shown  in  Fig.  23,  the  chronometer  being  placed  in  a  separate 


FIG.  23. 


circuit  with  a  single  cell,  connected  with  the  principal  circuit  by  a 
relay,  to  avoid  the  effects  of  too  strong  a  current  on  its  mechanism. 
The  chronometer  breaks  the  circuit  J,  releasing  the  armature  of  the 
chronometer  relay,  which  therefore  breaks  circuit  B  at  b.  This  re- 
leases the  armature  of  the  chronograph  magnet  to  which  is  attached 
a  pen,  thus  registering  on  the  chronograph  the  beats  of  the  chro- 
nometer. Circuit  B  may  also  be  broken  with  the  observing  key, 
thus  recording  the  transits  of  stars  also  on  the  chronograph.  At 
least  ten  well-determined  Ephemeris  stars  should  be  used — three 
equatorial  r.nd  two  circumpolar  for  each  position  of  the  transit. 


LONGITUDE. 


123 


Then  as  the  time  agreed  upon  for  the  exchange  of  signals  ap- 
proaches, the  local  circuit  should  be  connected  as  shown  in  Fig.  24, 

K 


(7,  chronometer  relay; 
Jf,  chronograph  magnet; 
K,  observing  key; 


FIG.  24. 
5,  sounder; 
L,  L,  main  line; 
K',  break-circuit  key; 


D,  relay; 

G,  galvanometer; 

R,  rheostat. 


by  a  relay  to  the  main  line,  which  is  worked  by  its  own  permanent 
batteries,,  and  in  which  there  is  also  a  break-circuit  key.  The  con- 
nections are  the  same  at  both  stations.  By  this  arrangement  it  is 
seen  that  each  chronograph  will  receive  the  time-record  of  its  own 
chronometer;  and  also  the  record  of  any  signals  sent  over  the  main 
line  in  either  direction. 

Neither  chronograph  receives  the  record  of  the  other's  chro- 
nometer. Then  at  the  time  agreed  upon,  warning  is  sent  by  the 
station  having  precedence,  and  the  signals  follow  according  to  any 
prearranged  system.  Notice  being  given  of  their  completion,  the 
second  station  signals  in  the  same  manner. 

As  an  example  of  a  system,  let  the  break-circuit  key  in  the  main 
line  be  pressed  for  2  or  3  seconds  once  in  about  ten  seconds,  but 
at  irregular  intervals :  this  being  continued  for  five  minutes  will 
give  31  arbitrary  signals  from  each  station. 

Each  chronometer  sheet  when  marked  with  the  date,  one  or 
more  references  to  actual  chronometer  time,  and  the  error  of 
chronometer,  as  soon  as  found,  will,  in  connection  with  the  sheet 
from  the  other  station,  afford  the  obvious  means  of  finding  all  the 
quantities  in  Eq.  (163)  from  which  the  longitude  is  computed. 
The  sheets  may  be  compared  by  telegraph,  if  desired. 

The  work  of  a  single  night  is  then  completed  by  transit  observa- 
tions upon  at  least  ten  more  stars  under  the  same  conditions  as 


124 


PRACTICAL  ASTRONOMY. 


before,  the  entire  series  of  twenty  being  so  reduced  as  to  give  the 
chronometer  error  at  the  middle  of  the  interval  occupied  in  ex- 
changing signals.  The  mode  of  making  this  reduction  will  be 
explained  hereafter. 

The  preceding  is  called  the  method  by  "  Arbitrary  Signals/'  and 
is  the  one  now  usually  employed.  Sometimes  however  the  method 
by  "Chronometer  Signals"  is  used,  which  will  be  readily  under- 
stood by  reference  to  Fig.  25,  the  connections  being  the  same  at 
both  stations. 


FIG.  25. 

In  this  case  it  is  seen  that  each  chronometer,  although  in  local 
•circuit,  graduates  each  chronograph,  upon  which  we  therefore  have 
a  direct  comparison  of  the  two  time-pieces. 

This  method  is  subject  to  the  inconvenience  and  possible  inac- 
curacies in  reading  which  may  occur  due  to  a  close  but  not  perfect 
coincidence  in  beats,  unless  special  precautions  are  taken. 

The  arrangement  of  the  galvanometer  and  rheostat,  as  shown  in 
both  figures  (taken  from  the  Coast  Survey  Keport  for  1880),  in- 
sures the  equality  of  the  currents  passing  through  the  relays  at  the 
two  stations,  which  point  should  be  ascertained  by  exchange  of 
telegraphic  messages;  therefore  after  the  relays  are  properly  ad- 
justed they  will  be  demagnetized  by  the  signals  with  equal  rapidity, 
and  constant  errors  in  this  respect  be  avoided. 

The  final  adopted  value  of  the  longitude  should  depend  upon 
the  results  of  at  least  five  or  six  nights;  outstanding  errors  in  the 
electrical  apparatus  being  nearly  eliminated  by  an  exchange  between 
the  two  stations  when  the  work  is  half  completed. 

"  Longitude  by  the  Electric  Telegraph  "  had  its  origin  in  the 


LONGITUDE.  125 

IT.  S.  Coast  Survey,  and  has  since  been  employed  considerably  in 
Europe.  As  at  first  employed  it  consisted  virtually  in  telegraphing 
to  a  western,  the  instant  of  a  fixed  star's  culmination  at  an  eastern 
station  ;  and  afterwards,  telegraphing  to  the  eastern,  at  the  instant 
of  the  same  star's  culmination  at  the  western  station. 

In  connection  with  Talcott's  Method  for  Latitude,  it  has  been 
used  extensively  in  important  Government  Surveys,  taking  prece- 
dence, whenever  available,  over  all  other  methods. 

Reduction  of  the  Time  Observations.  See  Form  12a.  —  These 
observations,  as  just  stated,  are  in  two  groups  ;  one  before,  and  one 
after  the  exchange  of  signals  or  comparison  of  chronometers.  From 
them  is  is  be  obtained  the  chronometer  error  at  the  epoch  of  ex- 
change or  comparison,  which  is  assumed  to  be  the  middle  of  the 
interval  consumed  in  the  exchange;  this  latter  being  about  12 
minutes. 

Let  us  resume  the  equation  of  the  Transit  Instrument  approxi- 
mately in  the  meridian, 


a=  T+  E+aA  +  lB+  C  (c  -  .021  cos  0),        (164) 

and  let  T0  denote  the  epoch,  or  the  known  chronometer  time  to 
which  the  observations  are  to  be  reduced.  Let  us  suppose  also, 
that  of  the  three  instrumental  errors,  a,  b,  and  c,  only  b  has  been 
determined,  this  being  found  directly  by  reading  the  level  for  every 
star.  The  rate  of  the  chronometer,  r,  is  supposed  to  be  known 
approximately,  and  it  is  to  be  borne  in  mind  that  E  is  the  error  at 
the  time  T.  Then  in  the  above  equation  E,  a,  and  c  are  un- 
known. 

Now  if  we  denote  the  error  at  the  epoch  by  EQ  ,  we  shall  have 

E  =  E0-(TQ~T)r.  (165) 

And  if  E'0  denote  an  assumed  approximate  value  of  E0  ,  and  e  be 
the  unknown  error  committed  by  this  assumption,  we  shall  have, 

E  =  E\+e-(T0-  T)  r,  (166) 

From  which,  Eq.  (164)  becomes 
e  +  Aa  +  Cc  +  T  -  .021  cos  0  0  +E\  -  (  T0  -  T)  r  +  El  -  a  =  0, 


126  PRACTICAL  ASTRONOMY. 

in  which  everything  is  known  save  e  (the  correction  to  be  applied 
to  the  assumed  chronometer  error  at  the  epoch),  a,  and  c. 

Aa  is  called  the  correction  .for  azimuth. 
Cc         "        "          "  "    collimation. 

—  .021  cos  0(7         "        "  "J         "    diurnal  aberration. 

—  (T0—T)r         "        "          "  "    rate. 

Bl         "        "  "  "    level. 

Collecting  the  known  terms,  transposing  them  to  the  2d  mem- 
ber, and  denoting  the  sum  by  n,  we  have 

e  +  Aa  -\-  Cc  =  n.  (167) 

Each  one  of  the  twenty  stars  furnishes  an  Equation  of  Condition 
of  this  form,  from  which,  by  the  principles  of  Least  Squares,  we 
form  the  three  "  Normal  Equations/' 

2  (C)  e  +  2  (A  C)  a  +  2  (0*)  c  =  2  (On), 


from  a  solution  of  which  we  find  a,  c,  and  the  correction,  e,  to  be 
applied  to  the  assumed  chronometer  error  at  the  epoch. 

If  either  c  or  a  be  known,  say  c,  by  methods  given  under  "  The 
Transit  Instrument,"  then  the  correction  for  collimation  for  each 
star,  C  c,  should  be  transferred  to  the  2d  member  and  included  in  n. 
We  then  have  only  the  two  "  Normal  Equations/' 


2  (A)  c  +  2  (A*)  a  =  2  (A  ri), 

from  which  to  find  6  and  a. 

It  is  to  be  remembered  that  the  middle  ten  stars  have  been  ob- 
served with  the  instrument  reversed,  and  that  such  reversal  changes 
the  sign  of  c,  and  therefore  of  the  term  C  c.  Hence  in  forming  the 
t(  Equations  of  Condition"  for  those  stars,  care  should  be  taken  to 
introduce  this  change  by  reversing  the  sign  of  C.  The  sign  of  c  as 


LONGITUDE.  127 

found  from  the  "  Normal  Equations  "  will  then  belong  to  the  col- 
limation  error  c  of  the  unreversed  instrument. 

Also,  since  reversing  the  instrument  almost  invariably  changes 
0,  it  is  better  to  write  a'  for  a  in  the  corresponding  "  Equations  of 
Condition/'  and  treat  a'  as  another  unknown  quantity.  We  will 
thus  have  four  "  Normal  Equations  "  instead  of  three,  and  derive 
from  them  two  values  of  the  azimuth  error,  one  for  each  position 
of  the  instrument. 

Sometimes,  and  perhaps  with  even  greater  accuracy,  the  solution 
is  modified  as  follows: 

Independent  determination  of  a  and  c  are  made,  as  explained 
heretofore,  by  the  use  of  three  stars. 

Adopting  these,  each  star  gives  a  value  of  the  chronometer  error 
as  per  Form  1.  The  mean  result  compared  with  the  similar  mean 
of  preceding  and  following  nights,  gives  the  rate.  The  principle  of 
Least  Squares  is  then  applied  (correcting  also  for  rate)  in  the  man- 
ner just  detailed,  to  obtain  the  corrections  to  be  applied  to  these 
values  of  a,  c,  and  the  mean  chronometer  error.  With  these  cor- 
rected values  of  a  and  c,  new  values  of  the  chronometer  errors  are 
found  by  direct  solution  (Form  1),  the  mean  of  which  is  adopted. 

»!«  Personal  Equation. — From  (163)  it  is  seen  that  although 
errors  in  E  and  E'  affect  the  deduced  longitude  directly,  the  effect 
will  disappear  if  they  are  equally  in  error. 

Practical  observers  acquire  as  a  rule  certain  fixed  habits  of  ob- 
servation whereby  the  transits  of  stars  are  recorded  habitually 
slightly  too  early  or  too  late,  thus  affecting  the  deduced  clock  error 
correspondingly. 

The  difference  between  the  result  obtained  by  any  observer  and 
the  true  value  is  called  his  Absolute  Personal  Equation,  and  that 
between  the  results  of  two  different  observers  their  Relative  Per- 
sonal Equation.  In  Longitude  work  this  latter  should  always  be 
determined  and  applied  to  one  of  the  clock  errors,  thus  giving 
values  of  E  and  E '  as  though  determined  by  a  single  observer,  and 
causing  them  if  in  error  at  all,  to  be  as  nearly  equally  so  as  possible. 

To  determine  this  Relative  Personal  Equation,  the  two  observers 
should,  both  before  and  after  the  longitude  work,  meet  arid  compare 
as  follows :  one  notes  the  transits  of  a  star  over  half  the  wires  of 
the  instrument,  and  the  other  the  transits  over  the  remaining  half. 
Each  time  of  transit  is  then  reduced  to  the  middle  wire  by  the 


128  PRACTICAL  ASTRONOMY. 

Equatorial  Intervals,  and  the  difference  between  their  respective 
means  will  be  a  value  of  their  relative  personal  equation.  The 
adopted  value  should  depend  upon  twenty  or  thirty  stars,  and  the 
work  be  distributed  over  three  or  four  nights. 

Personal  equation  is  not  a  constant  quantity,  and  should  be  re- 
determined  from  time  to  time.  On  j;he  Coast  Survey  it  is  largely 
eliminated  by  causing  the  observers  to  change  places  upon  comple- 
tion of  half  the  observations  for  difference  of  longitude  between  the 
stations. 

Application  of  Weights  and  Probable  Error  of  Final  Result.  — 
The  probable  error  of  an  observed  star  transit  may  be  divided  fdr 
practical  purposes  into  two  parts:  the  first,  due  to  errors  (apart 
from  personal  equation)  in  estimating  the  exact  instants  of  the 
star's  passage  over  the  wires,  unsteadiness  of  star,  etc.,  is  called  the 
observational  error;  the  second,  called  the  culmination  error,  is  due 
to  abnormal  atmospheric  displacement  of  star,  in  exact  determina- 
tion of  instrumental  errors,  anomalies  and  irregularities  in  the  clock 
rate,  etc.  Evidently  the  first  is  the  only  part  of  the  probable  error 
which  may  be  diminished  by  increasing  the  number  of  wires.  It 
may  be  determined  for  each  observer  as  follows  : 

Having  made  several  (m)  determinations  of  the  Equatorial  In- 
tervals as  before  explained,  let  each  be  compared  with  its  known 
value,  giving  for  the  probable  error  of  a  single  determination 
(Johnson,  Art.  72), 


.  /?<  =  0.6745  .  .  (a) 

Since  these  intervals  depend  upon  observed  transits  over  two  wires, 
we  have  for  the  probable  error  of  an  observed  transit  of  an  equato- 
rial star  over  a  single  wire  (Johnson,  Art.  87), 


For  any  other  star  this  will  manifestly  be 

R"  sec  8, 


LONGITUDE.  129 

and  for  N  wires  the  probable  error  of  the  mean  will  be 

nn 


For  the  smaller  instruments  of  the  Coast  Survey  R"  =  08.08  about. 
To  determine  the  culmination  error,  R',  for  an  equatorial  star, 
let  R  denote  the  combined  effect  of  both  errors;  then 


R  may  be  found  by  comparing  several  (m)  determinations  of  a 
star's  R.  A.  (all  reduced  to  the  same  equinox)  with  their  mean, 
using  the  same  formula  as  before.  Multiplying  the  value  thus 
found  by  cos  tf,  we  have  the  probable  error  for  an  equatorial  star. 
The  mean  result  from  many  stars  should  be  the  adopted  value  of  R. 

For  the  smaller  instruments  of  the  Coast  Survey  R  =  08.06 
about. 

Substituting  in  (c),  making  N  =  15, 

R'  =  05.056o 

For  any  other  star  this  will  evidently  be  R'  sec  8.  Hence  for  the 
probable  error  of  the  transit  of  an  equatorial  star  over  N9  or  the 
full  number  of  wires, 


and  for  any  less  number  of  wires, 

R,  = 


n 


130  PRACTICAL  ASTRONOMY. 

Since  the  weights  of  observations  are  proportional  to  reciprocals  of 
squares  of  probable  errors,  we  have  for  the  weight  of  an  observation 
on  n  wires  (that  011  the  full  number  being  taken  as  unity), 


0.0032.        +1 


0.003  + 

n  n 

Again,  from  what  precedes  it  is  seen  that  the  total  probable  error 
(R)  of  the  transit  of  an  equatorial  star  will  become  R  sec  d  for  any 
other.  Hence  different  stars  will  have  weights  inversely  as  seca  d. 
In  practice,  however,  slightly  diif  erent  relations  have  been  found  to 
answer  better.  For  the  instruments  above  referred  to,  the  formula 


,  _       JL.6  _ 
P    "1.6  -|-  tan2  d 

has  been  adopted. 

The  report  of  the  Chief  of  Engineers  for  1873  gives 

•  -L.O  /7  \ 

P  =  1  +  0.3  see'  f 

Therefore  if  each  Equation  of  Condition  in  the  Reduction  of  the 
Time  Observations  be  multiplied  by  the  corresponding  value  of  Vp 
(Johnson,  Art.  126),  it  will  be  weighted  for  missel  wires. 

In  the  same  way,  if  multiplied  by  Vp'  it  will  be  weighted  for 
declination.     It  is,  however,  unusual  to  weight  for  declination  when 


The  normal  equations  having  been  formed  from  the  weighted 
equations  of  condition  in  the  usual  manner,  their  solution  will  give 
the  chronometer  error  and  its  weight,  pe.  (Johnson,  Arts.  132,133.) 

The  probable  error  of  a  single  observation  is  then  found  by  the 
formula,  (Johnson  Art.  138), 


--,  (i) 

m  —  q* 


LONGITUDE.  131 

where  the  residuals,  v,  are  formed  from  the  m  weighted  equations 
of  condition,  and  q  is  the  number  of  normal  equations. 

The  probable  error  of  the  chronometer  correction  as  determined 
by  a  single  night's  work  will  then  be 


Similarly  we  obtain  pe'  for  the  weight  of  the  chronometer  correction 
at  the  other  station,  and  the  weight  to  be  assigned  to  the  resulting 
longitude,  from  the  relation  between  weights  and  probable  errors, 
will  be 


The  weighted  mean  longitude  as  the  result  of  m'  nights'  work  will 
then  be 


with  a  probable  error 


Circumstances  must,  however,  decide  as  to  the  relative  weights  to  be 
assigned  to  the  results  of  different  nights.  If  the  observations  have 
been  conducted  on  a  uniform  system,  it  will  perhaps  be  better  to 
give  them  all  equal  weight. 

3.  Longitude  by  Lunar  Culminations.— The  moon  has  a  rapid 
motion  in  Eight  Ascension.  If,  therefore,  we  can  find  the  local 
times  existing  on  two  meridians,  when  the  moon  had  a  certain 
R.  A.,  their  difference  of  longitude  becomes  known  from  this  differ- 
ence of  times. 

Determine  the  local  sidereal  time  of  transit  or  E.  A.  of  the 
moon's  bright  limb,  and  denote  it  by  «v 

From  pp.  385-392,  Ephemeris,  take  out  the  E.  A.  of  the  center 
at  the  nearest  Washington  culmination.  This  ±  the  Sidereal  Time 


132  PRACTICAL  ASTRONOMY. 

of  semi-diameter  crossing  the  meridian,  according  as  the  east  or  west 
limb  is  bright,  taken  from  same  page,  will  give  the  R.  A.  of  the 
bright  limb,  at  its  culmination  at  Washington.  Denote  this  by  <*w. 
Now  if  an  approximate  longitude  be  not  known,  which  will 
seldom  be  the  case,  one  may  be  established  as  follows :  Let  v  = 
moon's  change  in  R.  A.  for.  one  hour  of  longitude,  taken  from  same 
page  of  Ephemeris.  Then  upon  the  supposition  that  this  is  uni- 
form, we  will  have 

t;  :  1 : :  <r,  -  «w  :  £',    or    L'  =  °l  ~  "w, 


L9  being  the  approximate  longitude  from  "Washington,  whose 
longitude  from  Greenwich  is  accurately  known.  With  this  value 
of  L'  take  from  the  Ephemeris  a  new  value  of  v  corresponding  to 
the  mid-longitude  J  L',  and  determine  as  before  a  closer  approxi- 
mate longitude,  L".  If  we  are  within  two  hours  01  Washington 
in  longitude,  L"  will  be  sufficiently  close  for  the  purposes  to  which 
we  are  to  apply  it.  If  farther  away,  make  one  or  two  more  approx- 
imations, and  call  the  final  result  Lap. 

Lap  will  be  true  within  a  very  few  seconds  of  time  even  if  the 
observing  station  be  in  Alaska,  situated  6  hours  from  Washington, 
and  even  if  the  observations  be  made  when  the  moon's  irregularities 
in  R.  A.  are  most  marked. 

With  the  approximate  longitude  (and  this  is  one  of  the  uses  to 
be  made  of  this  quantity,  before  referred  to),  we  may  now  find  the 
sidereal  time  required  for  moon's  semi-diameter  to  cross  the  merid- 
ian of  the  place  of  observation  by  simple  interpolation  to  2d  or  3d 
differences  in  the  proper  column  of  the  same  page  of  the  Ephemeris. 
Denote  this  by  7\. 

The  greatest  change  in  the  time  required  for  semi-diameter  to 
cross  the  meridian,  due  to  a  change  of  one  hour  in  longitude,  is 
about  0.13  sec.  Hence,  even  if  we  could  possibly  have  made  an  error 
of  10  minutes  in  our  determination  of  L^p ,  the  value  of  Tt  can  only 
involve  an  error  of  about  .02 sec  when  at  its  maximum.  This  would 
involve  a  maximum  error  of  about  0.5sec  in  the  resulting  longitude. 

at±  TI  =  ofc  will  then  be  the  R.  A.  "of  the  moon's  center  at  the 
instant  of  transit  of  the  center. 

On  Pages  V  to  XII  of  the   Monthly  Calendar  are  found  the 


LONGITUDE.  133 

B.  A.  of  the  moon's  center  for  each  hour  of  Greenwich  mean  time. 
The  problem  now  is  to  find  at  what  instant  (  Tg)  of  Greenwich  time 
the  moon's  center  had  the  E.  A.  determined  by  our  observation. 
This  may  be  solved  by  an  inverse  interpolation;  i.e.,  instead  of 
interpolating  a  K.  A.  corresponding  to  a  given  time  not  in  the 
table,  we  are  to  interpolate  a  time  to  a  given  R.  A.  not  in  the  table  ; 
and  in  this  interpolation  the  use  of  second  differences  will  be  quite 
sufficient. 

Therefore  let  T0  and  T0  +  1  he  the  two  Greenwich  hours  be- 
tween which  a,  occurs. 

Let  d  a  be  the  increase  of  moon's  R.  A.  in  one  minute  of  mean 
time,  at  T0.  This  is  given  on  the  same  page. 

Let  6'  a  be  the  increase  of  S  a  in  one  hour.  Found  from  same 
column  by  subtracting  adjacent  values  of  d  a. 

Let  a0  be  the  R.  A.  given  in  the  Ephemeris  at  T0. 

Then  using  second  differences,  we  have 


In  this  equation  Tg  —  T0  is  expressed  in  seconds  ;  everything  is 
known  but  it,  and  its  value  may  be  found  by  a  solution  of  the 
quadratic.  The  result  added  to  T0  gives  Tg  ,  or  the  Greenwich  mean 
time  at  which  the  moon's  center  had  otc  for  its  R.  A.  Convert  this 
into  Greenwich  sidereal  time,  call  the  result  ag  ,  and  our  longitude 
is  known  from 

\=ag  —  are.  (169) 

The  preceding  is  the  method  to  be  followed  where  there  is  but 
a  single  station. 

Imperfections  in  the  Lunar  Tables  from  which  the  Ephemeris 
is  computed,  render  the  tabular  R.  A.  liable  to  slight  errors.  There- 
fore from  Equation  (168)  our  values  of  Tg  and  hence  ag  may  be 
incorrect  from  this  cause,  giving  from  Equation  (169)  an  incorrect 
longitude. 

Differences  between  two  tabular  values  are,  however,  nearly  cor- 
rect. 

Hence  it  is  more  accurate  to  have  corresponding  observations 


134  PRACTICAL  ASTRONOMY. 

of  the  moon's  transit  on  the  same  day  taken  at  a  station  whose 
longitude  is  known. 

Its  longitude,  found  as  above,  will  be 


and  the  difference  of  longitude  between  the  two  stations, 
V  -  A  =  «  -  ag)  -  (aa'  -  <*>), 

inaccuracies  of  the  Ephemeris  being  nearly  eliminated  in  the  differ- 
ence (<xg'  —  ag). 

No  method  of  determining  longitude  by  Lunar  Culminations  is 
sufficiently  accurate  for  a  fixed  observatory.  It  may  however  be 
used  in  surveys  and  expeditions  where  telegraphic  connection  with 
a  known  meridian  can  not  be  secured.  Even  with  the  appliances 
of  a  fixed  observatory,  the  mean  of  several  determinations  is  some- 
times subsequently  found  to  be  in  error  by  from  4  to  6  seconds  of 
time  (Madras  Observatory).  Dependence  should  not  therefore  be 
placed  upon  a  single  observation,  but  the  operation  should  be  re- 
peated upon  each  limb  as  many  times  as  may  seem  desirable.  The 
longitude  derived  from  any  determination  may  be  employed  as  the 
approximate  longitude  required  in  any  subsequent  determination. 

Before  proceeding  to  any  details  as  to  the  observations  and  re- 
ductions, it  is  well  to  note  the  effect  of  errors  in  either,  upon  our 
result.  The  main  outline  of  the  problem  consists  in  determining 
the  moon's  R.  A.  at  a  certain  instant,  and  then  ascertaining  from 
the  Ephemeris  the  Greenwich  time  of  the  same  instant.  Both  the 
moon's  R.  A.  and  the  instant  are  denoted,  at  the  place  of  observa- 
tion, by  ofc  =  a  i  ±  TI  .  txt  depends  very  largely  upon  accuracy  of 
observation  and  reduction.  Tl  depends  upon  interpolation  with  an 
approximate  longitude.  As  shown  before,  no  error  of  assumed  longi- 
tude that  could  ever  occur  in  practice,  would  have  any  appreciable 
effect  on  Tt.  If  the  interpolation  be  properly  performed,  Tl  can 
involve  only  very  slight  errors.  But  "whatever  they  may  be,  they 
enter  with  full  effect  in  ote  ,  and  when  the  final  operation  is  per- 
formed to  determine  the  corresponding  Greenwich  time,  an  inspec- 
tion of  the  tables  will  show  that  any  error  in  ac  is  increased  from 


LONGITUDE.  135 

£0  to  30  times  in  the  resulting  longitude.  In  this  way,  as  before 
shown,  an  error  of  .02s  in  Tl  is  amplified  into  .5s  in  the  result. 

Errors  in  al  affect  atc,  and  therefore  the  result,  in  the  same 
manner;  hence  we  see  that  considerable  care  is  necessary  in  both 
observation  and  reduction.  At  the  very  best,  the  result  is  liable  to 
be  in  error  from  1  to  3  seconds.  In  latitude  of  West  Point,  1 
second  of  time  =  1142  feet  in  longitude. 

Observations  and  Reductions. — The  transit  instrument  is  sup- 
posed to  be  pretty  accurately  adjusted  to  the  meridian,  and  the 
outstanding  small  errors  a,  b,  and  c,  measured.  The  rate  of  the 
sidereal  chronometer  is  also  supposed  to  be  known. 

Note  the  chronometer  time  of  transit  of  the  moon's  bright  limb 
over  each  wire  of  the  instrument.  In  this  case,  as  witji  a  star,  the 
time  of  culmination  is  found  by  reducing  the  observations  to  the 
middle  wire  and  then  correcting  for  the  three  instrumental  errors. 
See  Form  1.  But  in  case  of  the  moon  these  reductions  and  correc- 
tions take  a  somewhat  modified  form  due  to  the  two  facts  that  the 
moon  has  a  proper  motion  in  R.  A.,  and  also  a  very  sensible  parallax 
in  R.  A.  when  on  a  side  wire.  Hence  (see  note  following)  we  have 

F  instead  of sec  6',  for  the  reduction  to  the  middle  wire, 

n  n 

and  ( 4  a  +  B  b  -f  Cc')  Fcos  <?'  instead  of  A  a  -f  B  b  +  Cc,  for  the 
instrumental  correction;  and  the  Equation  of  the  Transit  Instru- 
ment as  applied  to  this  case  becomes, 

al=  —  +^-lF+fi  +  (Aa  +  Bb  +  Cc')  Fcos  3'.      (170) 

n          n 

In  this  equation  2  T  is  the  sum  of  the  observed  times,  n  the  num- 
ber of  wires  used,  2  i  the  sum  of  their  equatorial  intervals,  d'  the 
moon's  declination  as  seen,  i.e.,  as  affected  by  parallax,  and 


F  =  [1  -  p  sin  n  cos  (0'  -  *)]  sec  * 


p  being  the  earth's  radius  at  place  of  observation  in  terms  of  the 
equatorial  radius,  7t  the  moon's  equatorial  horizontal  parallax,  0' 
the  geocentric  latitude,  (da)  as  already  stated,  and  6  the  moon's 
geocentric  declination.  These  quantities  must  be  found  before  the 


136  PRACTICAL  ASTRONOMY. 

reduction  can  be  made.  The  mode  of  finding  p  and  0'  has  already 
been  explained.  To  find  TT,  d,  and  (d  a),  note  in  addition  to  the 
transit  of  the  moon's  limb  that  of  one  or  more  stars  at  about  the 
same  altitude,  and  which  culminate  within  a  few  minutes  of  the 
moon.  The  difference  between  the  times  of  passing  the  middle 
wire  applied  to  the  star's  R.  A.  will  give  an  approximate  value  of 
al9  from  which  an  approximate  longitude  is  determined  as  before 
explained.  With  this,  it  may  be  taken  from  page  IV,  and  d  and 
(d  a)  from  pp.  V  to  XII,  Monthly  Calendar.  F  thus  becomes 
known.  Evidently 

d'  —  6  —  />7rsin(0'  —  6) 

with  sufficient  accuracy,  and  the  computation  of  al  can  now  be 
made. 

One  of  the  greatest  inaccuracies  to  be  apprehended  is  a  failure 
to  determine  a  very  exact  value  of  E  for  the  instant  of  transit. 
This  quantity  may  be  eliminated,  or  very  nearly  so,  as  follows  : 

If  two  or  more  fundamental  stars,  those  whose  places  have  been 
established  with  the  highest  degree  of  accuracy,  be  selected  so  that 
the  mean  of  the  times  of  their  transits  shall  be  very  closely  the  time 
of  transit  of  the  moon's  limb,  then  the  mean  of  their  equations  will 
be,  corresponding  to  a  mean  star, 


Cc'}a.       (171) 


Subtracting  from  Eq.  (170),  since  E  and  Es  denote  errors  at 
almost  the  same  instant,  we  have 


(172) 


in  which  E  has  disappeared. 

If  E  and  Es  differ,  their  difference  will  be  simply  the  change  of 
error  in,  for  example,  ten  minutes,  which  can  be  accurately  allowed 
for  by  the  chronometer's  well-established  rate.  Moreover,  if  the 
stars  be  selected  so  that  their  declinations  differ  but  slightly  from 


LONGITUDE.  137 

that  of  the  moon,  it  is  evident  that  the  last  terms  of  Eqs.  (170)  and 
(171)  will  be  nearly  the  same,  and  that  their  difference  in  Eq.  (172) 
will  be  a  minimum.  See  expressions  for  A,  B,  and  C,  in  connection 
with  Form  1. 

By  this  method,  therefore,  the  E.  A.  of  the  moon's  limb,  at ,  is, 
from  Eq.  (172),  made  to  depend  very  largely  upon  the  R.  A.  of 
fundamental  stars;  instrumental  and  clock  errors  being  reduced  to 
a  minimum  of  effect. 

The  stars  should  be  selected  from  the  Ephemeris  in  accordance 
with  the  above  conditions,  and  observed  in  connection  with  the 
moon. 

>J*  To  deduce  Equation  (170). 

In  the  Equation   of  the  Transit   instrument,   the   quantities 

^"w*    * 

— -  sec  6  (embraced  in  T),  and  (Aa  +  Bb  -j-  Cc')  denote  respect- 
n 

ively  the  times  required  for  a  star  whose  declination  is  d  to  pass 
from  the  mean  to  the  middle  wire  and  from  the  middle  wire  to  the 
meridian.  In  the  case  of  the  moon  these  intervals  (or  hour-angles) 
require  modification,  both  on  account  of  parallax  and  proper  motion. 
The  Ephemeris  values  of  R.  A.  and  Declination  are  given  for 
an  observer  at  the  earth's  center;  but  on  account  of  our  proximity 
to  the  moon,  an  observer  on  the  surface  always  sees  that  body  dis- 
placed in  a  vertical  circle,  which  results  in  a  displacement  or  paral- 
lax both  in  declination  and  (unless  the  body  be  on  the  meridian) 
R.  A.  Hence  it  is  that  when  the  moon's  limb  appears  tangent  to 
a  side  wire  as  at  M' ,  Fig.  26,  it  is  in  reality  at  M.  Therefore  the 


FIG.  26. 


apparent  hour-angle  Z  P  M'  requires  a  correction  to  reduce  it  to 
the  true  hour-angle  Z  P  M?  and  the  result  is  to  be  further  modified 


138  PRACTICAL  ASTRONOMY. 

due  to  the  moon's  own  motion  in  R.  A.  The  following  is  based  on 
the  method  given  by  Chauvenet. 

To  deduce  the  relation  between  the  true  and  apparent  hour- 
angles,  let  them  be  represented  respectively  by  P  and  P',  the  cor- 
responding zenith  distances  by  z  and  z',  and  the  declinations  by  d 
and  6',  Z  being  the  geocentric  zenith. 

Then 

sin  P  :  sin  A  : :  sin  z  :  cos  #, 
sin  P'  N:  sin  A  : :  sin  zf  :  cos  6'9 
sin  P  sin  z  cos  d 


sin  P'  '     "  sin  z'  '  cos  6" 

.        .  sin  2    cos  6' 

sin  P  =  sin  P  — — > -nro 

sm  2'  cos  d 


Or,  since  P  and  P'  are  very  small  when  the  limb  is  on  a  side  wire, 
we  have,  expressing  them  both  in  seconds, 


p=p,smz_  cos 

sin  z'  cos 


P  is  the  time  which  the  limb  with  an  hour-angle  Pf  would  require 
to  reach  the  meridian  if  the  moon  had  no  proper  motion.  The 
actual  interval  is  greater  than  P  on  account  of  the  moon's  contin- 
ual motion  eastward  or  increase  in  R.  A.,  resulting  in  a  retardation 
of  its  apparent  diurnal  motion. 

To  determine  this,  the  Ephemeris  gives  at  intervals  of  one  hour 
the  moon's  motion  in  seconds  of  R.  A.  in  one  mean  solar  minute 
=  da.  One  m.  s.  minute  =  GO  X  1.002738  ^  60.1643  sidereal 

seconds.     Hence  in  one  sidereal  second  the  moon  moves 


ou.  - 
seconds  eastward,  and  therefore  its  apparent  diurnal  motion  west- 

Ok 

ward  .is  only  1  --  in  the  same  interval.      In  other  words, 

bO. 


this  is  the  apparent  rate  of  the  moon  in  diurnal  motion  at  the  in- 


LONGITUDE,  139 

stant  considered.     Denote  it  by  R.     Then  the  time  required  to 
traverse  the  true  hour  angle  P  (or  the  apparent,  P'),  will  be 

p,  sin  z  cos  &'  1 
sin  z'  cos  d  R' 


When  the  limb  is  on  the  mean  of  the  wires,  the  apparent  hour 

•%r»    • 

angle,  P',  from  the  middle  wire  becomes  -  -  sec  df  (since  d',  not  &,  is 

Ws 

the  declination  of  the  point  as  observed),  and  when  on  the  middle 
wire  P'  becomes-  [a  sin  (0  —  d')  -f  b  cos  (0  —  6')  -f  c'\  sec  6'. 

Hence  to  pass  from  the  mean  of  the  wires  to  the  meridian  re- 
quires 

.,      4    "  '"  •     >  •-»"'"'*• 

^  sec  d'  +  (a  sin  (0  -  tf')  +  0  cos  (0  -  <?')  +  (A  sec  d 


x  sin  z  cos  d'   1      2  i  sin  2      1 

x  5  =  ITS?  S 


cos 


Placing  -  — j  sec  d  —  =  F,  the  Equation  of  the  Transit  instru- 
sin  z  £& 

ment  as  applied  to  the  moon,  becomes,  designating  the  R.  A.  of  the 
limb  by  al9 


at  =  —  +  E  +  — l  F  +  (Aa  +  55  +  Cc')  Pcos  d'.    (170) 


For  purposes  of  computation  the  value  of  F  may  be  simplified 
by  expressing  — . — ,  in  terms  of  quantities  given  in  the  Ephemeris. 

Let  n  =  moon's  equatorial  horizontal  parallax,  p  the  parallax  in 
altitude,  and  p  as  heretofore. . 


\ 
140  PRACTICAL  ASTRONOMY. 

Then 

sin  z  _  sin  (zf  —  p)  _  sin  zf  cosp  —  cos  z'  smp  _ 
sin  z'  ~        sin  z'  sin  z' 

cosj?  —  cos  z'  p  sin  TT  ; 
since    sin  p  =  p  sin  n  sin  2'. 

Expanding  cos  z'  =  cos  (z  -}-p),  placing  sin'jt?  =  0  and  cos  z  = 
cos  (0'  —  tf),  we  have 

sin  z  .  ,    ,       .. 

— T  =  1  —  ft  Sin  7T  COS  (0    —  #) 

sins' 
and 

.F  =  [1  —  p  sin  TT  cos  (0'  —  <?)]  sec  tf  -^. 

Evidently  we  may  also  write, 

6'  =  3  -  pTtsin  (0'  -  6). 


4.  Longitude  by  Lunar  Distances.— On  pp.  XIII  to  XVIII  of  the 

Monthly  Calendar  in  the  Ephemeris  are  found  the  true  or  geocen- 
tric distances  of  the  moon's  center  from  certain  fixed  stars,  planets, 
and  the  sun's  center,  at  intervals  of  3  hours  Greenwich  mean  time. 
If  then  an  observer  on  any  other  meridian  determine  by  observation 
one  of  these  distances,  and  note  the  local  mean  time  at  the  instant, 
he  can  by  interpolation  determine  the  Greenwich  mean  time  when 
the  moon  had  this  distance,  and  hence  the  longitude  from  the 
difference  of  times. 

The  planets  employed  are  Venus,  Mars,  Jupiter,  and  Saturn,  and 
the  fixed  stars,  known  as  the  9  lunar-distance  stars,  are  a  Arietis 
(Hamal),  a  Tauri  (Aldebaran),  ft  Geminorum  (Pollux),  a  Loonis 
(Regulus),  a  Virginia  (Spica),  a  Scorpii  (Antares),  a  Aquilas 
(Altair),  a  Piscis  Australis  (Fomalhaut),  and  a  Pegasi  (Markab). 
From  this  list  the  object  is  so  selected  that  the  observed  distance 
shall  not  be  much  less  than  45°,  although  a  less  distance  may  be 
used  if  necessary. 

The  distance  observed  is  that  of  the  moon's  bright  limb  from  a 
star,  from  the  estimated  center  of  a  planet,  or  from  the  nearest 


LONGITUDE.  141 

limb  of  the  sun.     If  the  sextant  telescope  be  sufficiently  powerful 
to  give  a  well-defined  disc,  we  may  measure  to  the  nearest  limb  of 
the  planet,  and  treat  the  observation  as  in  the  case  of  the  sun. 
Thus  in  Fig.  27,  letting  Z  represent  the  observer's  zenith,  and  G' 


FIG.  27. 

and  C"  the  observed  places  of  the  sun  and  moon  respectively,  the 
distance  measured  is  S'  M',  from  limb  to  limb. 

The  effect  of  refraction  is  to  make  an  object  appear  too  high,  and 
that  of  parallax,  too  low.  In  the  case  of  the  sun  the  former  out- 
weighs the  latter.  In  the  case  of  the  moon  the  reverse .  is  true. 
Hence  the  tr-ue  or  geocentric  places  of  the  two  bodies  would  be 
represented  relatively  by  8  and  M,  and  the  distance  8  M,  from 
center  to  center,  is  the  one  desired. 

The  outline  of  the  method  is  as  follows : 

Having  measured  the  distance  8f  M',  and  corrected  it  for  the 
two  semi-diameters;  and  having  also  measured  the  altitudes  of  the 
two  lower  limbs  and  corrected  them  for  the  respective  semi-diame- 
ters, we  have  in  the  triangle  Z ' Cf  C"  the  three  sides  given,  from 
which  we  find  the  angle  at  Z.  Then  having  corrected  the  observed 
altitudes  for  refraction,  semi-diameter  and  parallax,  we  have  in  the 
triangle  Z  S  M,  two  sides  and  the  included  angle  Z,  to  compute  the 
opposite  side  S  M. 

Before  proceeding  to  the  more  definite  solution,  three  points 
should  be  noticed. 

1st.  The  semi-diameter  of  the  moon  as  seen  from  the  surface  of 
the  earth  is  greater  than  it  would  appear  if  measured  from  the 
center  of  the  earth,  due  to  its  less  distance.  Hence  C"  M'  is  an 


142  PRACTICAL  ASTRONOMY. 

"  augmented  semi-diameter"  and  must  be  treated  accordingly.    The 
augmentation  in  case  of  the  sun  is  insignificant. 

2d.  Since  refraction  increases  with  the  zenith  distance,  the  re- 
fractioji  for  the  center  of  the  sun  or  moon  will  be  greater  than  that 
for  the  upper  limb,  and  that  of  the  lower  limb  will  be  greater  than 
that  'of  the  center.     The  apparent  distance 
of  the  limbs  is  therefore  diminished,  and  the 
whole  disc,  instead  of  being  circular,  presents 
an  oval  figure,  whose  vertical  diameter  is  the 
least,  and  horizontal  diameter  the  greatest, 
as  shown  in  Fig.  28.     Therefore  if  c  d  denote 
the  direction  of  the  measured  distance,  the 
assumed  semi-diameter,  cf,  will  be  in  excess 
by  the  amount  e  /,  and  must  be  corrected 
accordingly.     This   correction   becomes  of 
FlG-  ^  importance  if  the  altitude  of  either  sun  or 

moon  be  less  than  50°  at  the  moment  of  observation. 

3d.  Since  the  vertical  line  at  the  station  does  not  in  general 
pass  through  the  earth's  center,  but  intersects  the  axis  at  a  point 
R.  (see  Fig.  17),  it  is  most  convenient  to  reduce  our  observations  at 
first  to  the  point  R,  regarding  the  earth  as  a  sphere  with  R  0  as  a 
radius,  and  then  to  apply  the  small  correction  due  to  the  distance 
C  Rf  in  order  to  pass  to  the  true  or  geocentric  quantities. 

In  the  following  explanation,  the  body  whose  distance  from  the 
moon  is  measured  is  taken  to  be  the  sun.  The  result  will  then 
apply  equally  to  a  planet  if  its  lirnb  be  considered ;  if  its  center  be 
considered,  the  expression  for  its  semi-diameter  becomes  zero.  If 
the  body  be  a  fixed  star,  the  expressions  for  its  semi-diameter  and 
parallax  become  zero. 
Let  h"  =  measured  altitude  of  moon's  lower  limb,  corrected  for 

sextant  errors. 

H"  =  measured  altitude  of  sun's  lower  limb,  also  corrected. 
d"  =  measured   distance   between   moon's   bright   limb   and 

nearest  limb  of  sun,  also  corrected. 
T    =  local  mean  solar  time  at  instant  of  measuring^". 
L'    =  an  assumed  approximate  longitude. 
0     =  latitude. 

Note  the  readings  of  the  barometer  and  of  the  attached  and  ex- 
ternal thermometers. 


LONGITUDE.  148 

With  T  and  L',  take  from  the  Ephemeris  the  following  quan- 
tities : 

s      =  geocentric  semi-diameter  of  moon. 

n     =  equatorial  horizontal  parallax  of  moon. 

$     =  geocentric  declination. 

8     =  semi-diameter  of  sun. 

D    =  geocentric  declination  of  sun. 

P    —  equatorial  horizontal  parallax  of  sun. 

i    The  first  two  are  obtained  from  page  IV,  monthly  calendar,  or 
pages  385  to  393  Sphemeris. 

The  third  from  pages  V  to  XII,  monthly  calendar,  or  pages  385 
to  393  Ephemeris. 

The  fourth  and  fifth  from  page  I,  monthly  calendar,  or  from 
pages  377  to  385  Ephemeris. 

The  sixth  from  page  278  Ephemeris. 

We  must  now  correct  d"  for  both  semi-diameters,  augmented  in 
case  of  the  moon.  Therefore  with  h"  +  s  and  s  as  arguments,  enter 
the  proper  table  and  take  out  the  amount  of  augmentation.  In  the 
absence  of  tables  this  may  be  computed  by  the  formula, 

Augmentation  =  Tc  s'  sin  (h"  +  s)  -f  I  k*  s*  +  \  k*  s3  sin2  (h"  +  s)  ; 

in  which  log  k  =  5.25020  —  10,  and  s  is  expressed  in  seconds.  (For 
deduction  of  this  series  see  Note  1.) 

Add  this  correction  to  s  and  we  have  s'  =  moon's  semi-diameter 
as  seen  from  point  of  observation. 

We  now  have  (neglecting  the  distortion  of  discs),  the  following 
values  of  the  observed  quantities  reduced  to  the  centers  of  the  ob- 
served bodies,  viz.  : 


Using  these  quantities  we  may  now  find  the  correction  due  to 
distortion  of  discs  (or  refractive  distortion),  as  follows:  From 
tables  of  mean  refraction  take  out  the  refractions  corresponding  to 
the  altitude  (h'  +  s')  of  the  upper  limb,  to  that  (hr  —  s')  of  the 
lower,  and  that  (h')  of  the  center.  The  difference  between  the 
latter  and  each  of  the  other  two  gives  very  nearly  the  contraction 
of  the  upper  and  lower  semi-diameters  of  the  moon.  This  may  be 
repeated  once  if  the  refractions  are  very  great  due  to  a  small  alti~ 


144  PRACTICAL  ASTRONOMY. 

tude.  The  mean  of  the  two  is  the  contraction  of  the  vertical 
semi-diameter  due  to  refraction.  Denote  it  by  AS,  and  the  same 
quantity  in  case  of  the  sun  by  A  S. 

These  quantities  are  represented  by  a  b  in  Fig.  28,  and  from  them 
we  are  to  find  ef,  or  the  distortion  in  the  direction  of  d" .  This  is 
found  to  vary  very  nearly  as  cos2  q, \"q  being  the  angle  which  d" 
makes  with  the  vertical.  (See  Note  2.) 

The  values  of  q,  or  Q  in  case  of  the  sun,  will  be  found  from  the 
three  sides  of  the  triangle  Z  C'  C'',  Fig.  27.  Their  values,  page  6, 
Book  of  Formulas,  will  be,  if  m  =  %  (d'+  h'  +  H'). 

cos  m  sin  (m  —  H')  .   ~       cos  m  sin  (m  —  h') 

sm3  \  q  =  -   —. — jf 77 — '-,    sm2 1  0  = r     ,,.  v — ^7 — '-. 

sin  d'  cos  h'  sm  d'  cos  H' 

And  the  refractive  distortions  will  be,  from  the  above, 
A  ,9  cos2  g,  and   A  8  cos2  Q. 

Hence  the  fully  corrected  values  of  the  measured  quantities  are 
d'  =  d"  +  (*'  -As  cos2  q)  +  (S  -  A£  cos2  Q), 
V  =  h"  +  sr  -  AS,  H9  =  H"  +  S  -  AS. 

We  now  have  the  distance  (d'),  between  the  centers  and  the 
altitudes  of  the  centers  (hf  and  H'),  as  these  quantities  would  have 
been  had  we  been  able  to  measure  them  directly.  We  must  now 
ascertain  what  they  would  have  been  had  we  measured  them  at  the 
center  of  the  earth;  or,  as  a  first  step,  had  we  measured  them  at  the 
point  R. 

This  is  necessary,  because  the  earth  not  being  a  perfect  sphere, 
the  transference  of  an  observer  to  the  center  would  not  displace  a 
body  (apparently)  toward  the  astronomical,  but  toward  the  geocen- 
tric zenith,  and  the  angle  at  Z,  Fig.  27,  would  no  longer  be  com- 
mon to  the  two  triangles.  But  by  regarding  the  earth  as  a  sphere 
with  radius  0  R,  Fig.  17,  the  two  zeniths  will  coincide,  and  the 
reduction  therefore  be  easily  made.  Afterward  a  correction  is  to  be 
applied  due  to  a  transference  of  the  observer  from  R  to  C. 

Therefore  let  Ht,  h,,  and  dt,  be  the  values  of  H',  h',  and  d', 
when  referred  to  R,  and  let  rt  and  r  be  the  actual  refractions  for 


LONGITUDE. 


145 


H'  and  h'.     It  will  be  shown  in  Note  3  that  n  (the  angle  sub- 
tended by  the  equatorial  radius  at  the  distance  of  the  moon)  is  to 

nl9  the  angle  subtended  by  0  R,  as  a,  the  equatorial  radius,  is  to  — . 

Therefore  it,  the  parallax  at  R,  equals  — .    On  account  of  the  greater 

distance  of  the  sun,  P  will  be  practically  the  same  for  R  as  for  C. 
Therefore,  Art.  83,  Young, 

ht  =  li'  —  r  +  K4  cos  (h'  —  r) 
#,  =  #'- r,  +  P  cos  (ff'-r,) 
In  order  to  find  dt,  let  S  and  M  (Pig.  29)  represent  'the  places 


FIG.  29. 


of  the  sun  and  moon  as  seen  from  the  point  R  without  refraction, 
given  by  H,,  h,,  and  d,\  and  C'  and  C"  the  places  as  observed, 
(given  by  H',  V  and  d'). 
Then  in  triangle  Z  C'  C", 

„      cos  d'  —  sin  h'  sin  //'      _,  „     . 

cos  ^  =  -         —  r, rr/ •     Page  6,  Book  of  Formulas. 

COS  iv    COS  .Li 


146  PRACTICAL  ASTRONOMY. 

From  triangle  Z  S  M, 

cos  d.  —  sin  li  .  sin  H. 

cos  Z  =  -  —  -  —  7  --  -FJ  --  '.     Page  6,  Book  of  Formulas. 
cos  lit  cos  Ht 

Equating  these  two  values  of  cosffe',  adding  unity  to  both  mem- 
bers and  reducing, 

cos  d'  +  cos  (V  -f  H')  =-cos  ^  +  cos  (ft,  +  #,) 
cos  li'  cos  77'  cos  7^  cos  Ht 

Make  m  =  £  (7*'+77'-M')>  whence  cos  (7*'-|-  77')  =  cos  (2  ra-d'). 
Substituting  in  the  preceding  equation,  reducing  the  first  member 
by  formulas  4,  page  4,  11,  page  2,  and  13,  page  1,  and  the  second 
member  by  formulas  9  and  10,  page  2,  we  have 


cos  m  cos  (m  —  d')  __  cos2  J  (]it  -\-  H4]  —  sin2  £  dt 
cos  A'  cos  Hf  cos  /£  cos 


Whence, 


1/7     ,    7-7-x      cos/iy  cos  Ff. 

sin2  1  ^y  =  cos2  i  (7^   +  ZT.)  ---  ^  --  ^  cos  m  cos  (m  —  ^'). 

cos  h  cos  ^T' 

/ 
This  'may  be  placed  in  a  more  convenient  form  by  assuming 

cos  Jit  cos  Ht    cos  m  cos  (m  —  d')  _    .  2 
cos  A'  cos  #'      cos2 


Whence  sin  J  ^y  =  cos  J  (^y  -J-  Ht)  cos  J^". 

We  now  have  the  distance  between  the  centers  as  it  would  have 
been  without  refraction,  if  measured  from  the  point  7£.  This  is 
represented  by  the  line  S  M.  (Fig.  29.) 

The  transference  of  the  observer  "to  the  center  will,  since  this 
motion  lies  wholly  in  the  plane  P  MR,  have  the  effect  of  appar- 
ently diminishing  the  declination  of  the  moon,  causing  it  to  appear 
at  M'  ',  while  the  position  of  S  will  not  be  sensibly  changed. 


LONGITUDE.  147 

It  will  be  shown  in  Note  4  that  the  correction  to  be  added  to 
d,  (S  M)  to  give  d  (8  Mr)  is 

7t  6*  sin0        /sinD  _  sin  fl\  _      .  /sinZ)  _  sin  #\ 
4/1  —  e*  sin2^   \sind,      tan^J  \sm^,      tandj* 

e  being  the  eccentricity  of  the  meridian  =  0.0816967. 

Hence  we  have  finally,  denoting  the  geocentric  distance  between 
centers  by  d, 


.  /sin  D       smd\ 

—  d.  +  n%  I-T — 7 r—j    • 

'   '         \sm  d,      tan  d,/ 


This  operation  of  finding  d  from  the  observed  quantities  is  called 
"  Clearing  the  Distance." 

It  is  now  necessary  to  find  the  Greenwich  mean  time  when  the 
moon  and  sun  were  separated  by  the  distance  d.  For  this  purpose 
enter  the  Ephemeris  at  the  pages  before  referred  to,  and  find  there- 
in two  distances  between  which  d  falls.  Take  out  the  nearer  of 
these  and  the  Greenwich  hours  at  the  head  of  the  same  column. 
Then  if  A  denote  the  difference  between  the  two  distances,  and  A  ' 
the  difference  between  the  nearer  one  and  d,  both  in  seconds,  we 
shall  have,  using  only  first  differences,  for  the  correction,  t,  to  be 
applied  to  the  tabular  time  taken  out, 

A:3h::  A':*h.-.*h  =  -  A'. 


3h 
Or  log  th  =  log  --  h  log  A  '. 

Or  in  seconds,         log  ts  —  log  =  --  j-  log  A  '. 

The  logarithms  of  -  are  given  in  the  columns  headed  "  P. 

L.  of  Diff."  (Proportional  Logarithm  of  Difference.)  Hence  we 
have  simply  to  add  the  common  logarithm  of  A  '  in  seconds  to  the 
proportional  logarithm  of  the  table  to  obtain  the  common  logarithm 
of  the  correction  in  seconds  of  time. 


148  PRACTICAL  ASTRONOMY 

To  take  account  of  second  differences,  take  half  the  difference 
between  the  preceding  and  following  proportional  logarithms. 
With  this  and  t  as  arguments  enter  table  1,  Appendix  to  Ephem- 
eris,  and  take  out  the  corresponding  seconds,  which  are  to  be  added 
to  the  time  before  found  when  the  proportional  logarithms  are 
decreasing,  and  subtracted  when  they  are  increasing. 

Denote  the  final  result  by  Tg ,  and  the  difference  of  longitude 
by  A.  Then 

\=Tg-T.  (173) 

The  mode  given  above  for  clearing  the  distance  is  quite  exact, 
but  somewhat  laborious.  There  are,  however,  several  approximative 
solutions,  readily  understood  from  the  foregoing,  which  may  be 
employed  where  an  accurate  result  is  not  required,  and  which  may 
be  found  in  any  work  on  Navigation. 

The  method  by  "  Lunar  Distances  "  is  of  great  use  in  long  voy- 
ages at  sea  or  in  expeditions  by  land,  where  no  meridian  instru- 
ments are  available,  and  when  the  rate  of  the  chronometers  can  no 
longer  be  relied  upon. 

It  is  important  to  note  that  if  Tin  Eq.  (173),  denote  the  chro- 
nometer time  of  observation,  instead  of  the  true  local  time,  Tg  —  T 
will  be  the  error  of  the  chronometer  on  Greenwich  time.  In  this 
way  chronometers  may  be  "  checked."  If,  however,  T  denote  the 
true  local  time,  obtained  by  applying  the  error  on  local  time  to  the 
chronometer  time,  then  the  same  equation  gives  the  longitude. 

Observations.— It  is  necessary  that  A",  //",  and  d"  should  cor- 
respond to  the  same  instant  T.  Hence  observe  the  following  order 
in  making  observations.  Take  an  altitude  of  the  sun's  limb,  then 
an  altitude  of  the  moon's  limb,  then  the  distance,  carefully  noting 
the  time,  then  an  altitude  of  the  moon's  limb,  then  an  altitude  of 
the  sun's  limb.  A  mean  of  the  respective  altitudes  of  the  two  limbs 
will  give  very  nearly  the  altitudes  at  the  instant  of  measuring  the 
distance. 

For  greater  accuracy,  several  measurements  of  the  distance  may 
be  made,  and  the  mean  adopted.  Also,  when  possible,  at  least  two 
stars  should  be  used  on  opposite  sides,  of  the  moon,  for  the  purpose 
of  eliminating  instrumental  errors. 

The  accuracy  of  the  result  will  depend  upon  the  observer's  skill 
with  the  sextant,  and  mode  of  reduction  followed. 


LONGITUDE.  149 

»f«  1.  To  Find  Augmentation  of  Moon's  Semi-diameter.— In  de- 
termining the  augmentation  of  the  moon's  semi-diameter  due  to  its 
altitude,  the  ellipticity  of  the  earth  is  practically  insensible.  There- 
fore (Young,  p.  62),  denoting  the  altitude  of  the  center  (h"  -\-  s) 
by  h'9  the  parallax  in  altitude  by  p,  and  the  augmented  semi-diameter 
by  s', 


?'  =  * 


COS  ll' 


cos 


,  .  ,  /  cos  li'  —  cos  (lir 

Augmentation  =  G  =  s'  —  s  =  s  \  -  /7,    v 

\        cosh' 


By  page  4,  Book  of  Formulas, 

cos  h'  —  cos  (h'  +  p)  =  2  sin  J  (h'  +  h'  +p)  sin 
2  s 

G  =  --  Tyy—  —  r  Sm  (ll'  +  |  »)  SHI  \  ffl. 

cos  ' 


Expanding  the  sine  and  cosine  of  the  sums,  writing  \  p  for  sin 
9,  and  unity  for  cos  J  p,  we  have 


cos  7^ 


cos  7^'  —  «  sin  li 


p  =  TT  cos^'  (Young,  p.  61). 

According  to  the  Tables  of  the  Moon,  the  relation  between  n 
and  s  is  constant,  such  that 


-  3.6697  s. 


Hence  ^  =  3.6697  s  cos  //. 
Designating  this  numeral  by  k' 


_  k'  s*  (sin  h'  +  \  V  s  cos2 
1  —  &'ssin  h' 


By  division, 

G  =  k's>  sin  A'  +  $  &"  5s  +  J-  ^'2  s"  sin*  /*'  +  etc. 


150  PRACTICAL  ASTRONOMY. 

Multiplying  by  sin  1"  to  reduce  G  to  seconds, 

G  =  Jc  s2  sin  h'  +  £  k*  ss  +  \  k'  s3  sin2  h'  +  etc.,         (174) 

in  which  log  k  —  5.2502  —.10. 

•{«  2.  To  Deduce  the  Law  of  Refractive  Distortion.  —  In  Fig.  28, 
let  A'  denote  the  altitude  of  the  center,  and  h"  that  of  any  point  of 
the  limb,  as  /.  Then  the  difference  of  mean  refraction  for  c  and  / 
will  be  (Young,  p.  64), 

coU"),  (1) 


in  which  a  is  the  constant  60".6. 

Denoting  the  angle  acfloyq,  and  the  semi-diameter  by  s, 

h"  =  h'  +  s  cos  q.  (2) 

From  Trigonometry, 

,,  „  _  1  —  tan  h'  tan  (s  cos  q) 

tan  h'  -j-  tan  (s  cos  q)  * 

Substituting  in  (1),  and  writing  s  cos  q  tan  1"  for  tan  (s  cos  q), 
we  have, 


&  _     (s  cos  #  *an  1 
\        tan3  h' 


tan9 


-f-  ^  cos  <7  tan  1"  tan  A'      / 

The  last  term  in  the  denominator  is  insignificant  compared  with 
tan8  h'  ;  hence 

F  =  a  cosec2  h'  s  cos  q  tan  1",  (3) 

which  by  (1)  and  (2)  will  be  the  difference  between  that  ordinate 
of  the  ellipse  and  the  circle  which  passes  through/. 
Hence  the  line  e/will  be, 

Kefractive  Distortion  =  a  cosec1  h'  s  tan  1"  cos*  q. 


LONGITUDE.  151 

If  q  —  0,  we  have  a  b  =  contraction  of  vertical  semi-diameter  = 
As  =  a  cosec2  Ji'  s  tan  1".     Hence  finally, 

Refractive  Distortion  =  A  s  cos2  q.  (175) 

»|«  3.  To  Deduce  the  Parallax  for  the  Point  -R. 
By  making  x  =  o  in  Equation  (95),  and  reducing  by  (108),  (99) 
and  (100),  we  have  for  the  distance  R  C  (Fig.  17  or  29), 

a  e*  sin  0 
Vl  —  e*  sin2  0* 

Denoting  the  distance  R  0  by  y,  the  triangle  ROC  gives 
ae2sin0  .    , 

"^  =r7~.  ri-:  #::  sm  (0  -  0  )  :  cos  0  • 

VI  —  e  sm'  0 

a  a1  sin  0  cos  0' 


~  sin  (0  -  0')  yT-  e'sin"  0* 
Developing  sin  (0  —  0'),  cancelling  and  applying  (125), 


Vl  -  e>  sin2  0 

Comparing  with  second  part  of  (112)  it  is  seen  that  the  denom- 
inator is  sensibly  the  value  of  p  expressed  in  terms  of  a  as  unity. 
Hence 


The  angles  at  the  moon  subtended  bv  the  two  lines  a  and  — 
will  be  proportional  to  those  lines.     Therefore 


n,  =  5.  (176) 

^*  4.  To  Determine  the  Difference  between  dt  and  dy  due  to  a 
Transference  of  the  Observer  from  R  to  C. 


152  PRACTICAL  ASTRONOMY. 

By  the  previous  note  we  have  (Fig.  29), 


Vl  -  e2  sin3  0 

The  perpendicular  distance,  C  N^  from  the  center  to  the  line 
R  M,  is,  with  an  error  entirely  negligible, 

• 
a  e*  sin  0  cos  d 


'  Vl  —  e2  sin3  0* 
As  before,  the  angle  at  the  moon  subtended  by  this  line  will  be 

a  e*  sin  0  cos  #  n  _  7t  e*  sin  0  cos  d 
Vl  —  e*  siny  0  a        V I  —  e'2  sin2  0 

which  is  therefore  the  angular  apparent  displacement  of  the  moon, 
represented  by  the  arc  M  M'  (Fig.  29). 

Denote  it  by  m.     Then,  in  the  triangles  P  M'  8  and  MM'  8, 


cos       ,as 


cos  d,  —  cos  m  cos  d       sin  D  —  ~  sin  tf  cos  d 
--  '-  --  —  ---      -         —-  —  - 


sin  m  sin  d  cos  6  sin  d 

Eeducing,  replacing  cos  m  by  unity, 
cos  d,  —  cos  d 


=  sin 


/sin  D       sin  tf  cos  d\ 


Vcostf 


cos  d     J 


FIG.  30. 


From  Fig.  30,  it  is  seen  that  when 
dt  and  d  are  nearly  equal,  as  in  the 
present  case,  we  may  replace  cos  d'  — 
cos  d  by  sin  (d  —  d4)  sin  dt . 

Therefore 


Or 


.  .         .         /     sin  D  sin  6  cos  d\ 

sin  (a  —  a.)  =  sin  m -- ^—. — -j ---- >— : — ^  ). 

\cos  d  sm  d,      cos  tf  sin  «,/ 

=  _^£sm_^=  /sin/)  _  rin*  y 
Vl  —  e*  sin2  0  xsin  rf,      tan  rf,/ 


OTHER  METHODS  OF  DETERMINING  LONGITUDES.      153 


OTHER   METHODS   OF   DETERMINING  LONGITUDE. 

1st.  If  two  stations  are  so  near  each  other  that  a  signal  made  at 
either,  or  at  an  intermediate  point,  can  be  observed  at  both,  the 
time  may  be  noted  simultaneously  by  the  chronometers  at  the  two 
stations,  and  the  difference  of  longitude  thus  deduced.  An  appli- 
cation of  the  same  system,  by  means  of  a  connected  chain  of  signal 
stations,  will  give  the  difference  of  longitude  between  two  remote 
stations.  The  signals  are  usually  flashes  of  light — either  reflected 
sunlight  or  the  electric  light,  passed  through  a  suitable  lens. 

3d.  By  noting  the  time  of  beginning  or  ending  of  a  lunar  or 
solar  eclipse,  or  by  occupations  of  stars  by  the  moon.  For  these 
methods,  see  various  Treatises  on  Astronomy. 

3d.  By  Jupiter's  Satellites.  a.  Prom  their  eclipses.  The 
Washington  mean-times  of  the  disappearance  of  each  'satellite  in 
the  shadow  of  the  planet,  and  reappearance  of  the  same,  are  accu- 
rately given  in  the  Ephemeris,  pp.452-473,  accompanied  by  diagrams 
of  configuration  for  convenience  of  reference.  A  full  explanation 
of  the  diagrams  is  given  on  p.  449.  An  observer  who  has  noted  one 
of  these  events,  has  only  to  take  the  diiference  between  his  own 
local  time  of  observation  and  that  given  in  the  Ephemeris,  to  obtain 
his  longitude.  This  method  is  defective,  since  a  satellite  has  a  sen- 
sible diameter  and  does  not  disappear  or  reappear  instantaneously. 
The  more  powerful  the  telescope  employed,  the  longer  will  it  con- 
tinue to  show  the  satellite  after  the  first  perceptible  loss  of  light. 
These  facts  give  rise  to  discrepancies  between  the  results  of  differ- 
ent observers,  and  even  between  those  of  the  same  observer  with 
different  instruments.  Both  the  disappearance  and  reappearance 
should  therefore  be  noted  by  the  same  person  with  the  same  instru- 
ment, and  a  mean  of  the  results  adopted.  The  first  satellite  is  to 
be  preferred,  as  its  eclipses  occur  more  frequently  and  more  sud- 
denly, although  both  disappearance  and  reappearance  cannot  be 
observed. 

b.  From  their  occultations  by  the  body  of  the  planet.  The  times 
of  disappearance  and  reappearance  to  the  nearest  minute  only,  are 
given  on  same  pages  of  the  Ephemeris.  Since  the  times  are  only 
approximate,  they  simply  serve  to  enable  two  observers  on  different 


154  PRACTICAL  ASTRONOMY. 

meridians  to  direct  their  attention  to  the  phenomenon  at  the  proper 
moment.  A  comparison  of  their  times  will  then  give  their  relative 
longitude. 

c.  From  their  transits  over  Jupiter's  disc. 

d.  From  the  transits  of  their  shadows  over  Jupiter's  disc.     The 
approximate  times  of  ingress  and  egress,  to  be  used  as  in  case  &,  are 
given  on  same  pages  of  the  Ephemeris,  for  cases  c  and  d. 

Application  to  Explorations  and  Surveys. — On  explorations,  and 
reconnoissances  for  more  exact  surveys,  the  observer  will  usually  be 
provided  only  with  a  chronometer,  sextant,  and  artificial  horizon, 
with  probably  the  usual  meteorological  instruments. 

The  chronometer  should  be  carefully  rated  and  have  its  error  on 
the  local  time  of  some  comparison  meridian  (e.g.,  that  of  Washing- 
ton) accurately  determined  for  some  given  instant,  so  that,  by  ap- 
plying the  rate,  its  error  on  the  same  local  time  may  be  found 
whenever  desired. 

The  sextant  should  have  its  eccentricity  determined  before 
starting,  since  this  error  often  exceeds  any  ordinary  index  error, 
and  cannot  be  eliminated  by  adjustment. 

The  observer  should  be  able  to  recognize  by  name  several  of  the 
principal  Ephemeris  stars.  To  determine  the  coordinates  of  his 
station  when  they  are  entirely  unknown,  he  should  first  find  the 
chronometer  error  on  his  own  local  time,  using  preferably  the 
method  by  "  equal  altitudes  of  a  star,"  since,  as  has  been  seen,  he 
will  then  be  independent  of  any  knowledge  of  the  star's  declination, 
his  own  time,  latitude,  longitude,  or  instrumental  errors. 

Observations  for  latitude  may  be  made  at  any  convenient  time 
by  "  circum-meridian  altitudes  "  of  a  south  and  north  star,  or  of  a 
south  star  only,  combined  with  "  Polaris  off  the  meridian/'  the 
reductions  being  made  by  aid  of  the  chronometer  error  just  re- 
ferred to. 

The  method  by  "  circumpolars  "  may  also  be  used  as  a  verifi- 
cation when  applicable,  the  reduction  being  very  simple. 

The  longitude  is  known  as  soon  as  the  chronometer  error  on 
local  time  is  known,  by  comparing  this  with  its  known  error  on  the 
local  time  of  the  comparison  meridian.  -  However  large  the  rate  of 
a  chronometer,  it  should  be  nearly  constant;  but  after  some  time 
spent  in  traveling,  with  possible  exposure  to  extremes  of  tempera- 
ture, its  indications  of  the  comparison  meridian  time  are  rendered 


OTHER  METHODS  OF  DETERMINING  LONGITUDES.      155 

somewhat  uncertain  by  the  accumulation  of  unknown  errors,  thus 
introducing  the  same  uncertainties  into  our  longitudes.  In  such 
cases  the  method  by  "  lunar  distances  "  will  afford  an  approximate 
reestablishment  of  the  chronometer  error  on  the  comparison  merid- 
ian time,  or  a  correction  to  an  assumed  approximate  longitude. 

If  it  be  impracticable  to  find  the  local  time  by  equal  altitudes  as 
recommended,  on  account  of  clouds  or  the  length  of  time  involved, 
it  may  be  found  by  "  single  altitudes  "  of  an  east  and  a  west  star 
(or  of  a  single  star  when  necessary,  either  east  or  west),  an  approxi- 
mate value  of  the  latitude  required  in  the  computation  being  found 
from  the  best  obtainable  value  of  the  meridian  altitude  of  the  star 
observed  for  latitude.  With  the  error  thus  found  the  latitude  is 
found  as  before,  which,  if  it  differs  materially  from  the  assumed 
approximate  value,  must  be  used  in  a  recomputation  of  the  time. 
From  this  the  longitude  follows  as  before. 

If  the  latitude  be  known  or  approximately  so,  as  at  a  fixed  sta- 
tion or  when  tracing  a  parallel  of  latitude,  time  and  longitude  will 
be  most  expeditiously  determined  by  "single  altitudes." 

In  certain  classes  of  work  it  is  necessary  to  obtain  approximate 
coordinates  by  day,  in  which  case  of  course  the  sun  must  be  used  in 
accordance  with  the  same  general  principles  as  far  as  applicable. 

In  all  sextant  work,  except  in  methods  by  equal  altitudes,  its 
adjustments  and  errors  must  be  carefully  attended  to. 

In  extensivejsurveys  and  geodetic  work,  where  very  precise  results 
are  required,  the  methods  employed  are  "  Time  by  Meridian  Tran- 
sits "  with  the  reduction  by  Least  Squares,  Longitude  by  the  Elec- 
tric Telegraph,  and  Latitude  by  the  Zenith  Telescope.  The  ob- 
serving-instruments  should  be  mounted  on  small  masonry  piers  or 
wooden  posts  set  about  four  feet  in  the  earth  and  isolated  from  the 
surrounding  surface  by  a  narrow  circular  trench  one  or  two  feet 
deep. 

The  exact  location  of  an  astronomical  station  is  preserved,  if  de- 
sired (as  when  the  station  is  one  extremity  of  a  base-line),  by  a 
cross  on  a  copper  bolt  set  in  a  block  of  stone  embedded  two  or  three 
feet  below  the  surface,  the  exact  location  of  which  is  recorded  by 
suitable  references  to  surrounding  permanent  objects. 

Often  it  is  required  to  determine  the  coordinates  of  a  point 
where  it  is  impracticable  to  locate  an  astronomical  station,  as  for 
example  alight-house  or  a  central  and  prominent  building  of  a  city. 


156  PRACTICAL  ASTRONOMY. 

In  such  a  case,  having  made  the  requisite  observations  at  a  suitable 
station  in  the  vicinity,  and  having  computed  by  (111)  and  (114)  the 
length  in  feet  of  one  second  in  latitude  and  longitude,  measure  the 
true  bearing  and  distance  of  the  point  from  the  station,  from  which 
the  coordinates  of  the  former  with  respect  to  the  latter  are  readily 
computed. 

In  locating  points  at  intervals  on  a  line  which  coincides  with  a 
parallel  of  latitude,  sextant  observations  for  latitude  which  can  be 
quickly  reduced  will  give,  as  just  explained,  the  approximate  dis- 
tance of  the  observer  from  the  desired  parallel,  to  the  immediate 
vicinity  of  which  he  is  thus  enabled  to  proceed.  At  this  point  a 
complete  series  of  observations  for  latitude  is  made  with  the  zenith 
telescope,  and  the  resulting  distance  to  the  parallel  carefully  laid 
off  due  north  or  south. 

In  this  manner  points  about  twenty  miles  apart  were  located  on 
the  49th  parallel  between  the  U.  S.  and  the  British  Possessions. 

TIME   OF   CONJUNCTION   OR  OPPOSITION. 

Two  celestial  bodies  are  said  to  be  in  conjunction  when  either 
their  longitudes  or  their  right  ascensions  are  equal;  and  in  opposition 
when  they  differ  by  180°.  In  the  Ephemeris  the  conjunctions  and  op- 
positions of  the  moon  or  planets  with  respect  to  the  sun  refer  to  their 
longitudes.  Conjunctions  of  the  moon  and  planets  or  of  the  planets 
with  each  other  refer  to  their  right  ascensions.  In  other  cases,  when 
used  without  qualification,  the  terms  usually  refer  to  longitudes. 

The  longitudes  of  the  principal  bodies  of  the  solar  system  (or 
the  data  from  which  they  may  be  computed)  are  given  in  the 
Ephemeris  for  (usually)  each  Greenwich  mean  moon.  To  find  the 
time  of  conjunction,  determine  by  inspection  of  the  tables  the  two 
dates  between  which  the  longitudes  of  the  bodies  become  equal,  and 
denote  the  earlier  date  by  T.  Take  from  the  tables  four  consecutive 
longitudes  for  each  body — two  next  preceding  and  two  next  follow- 
ing the  time  of  conjunction.  Form  for  each  the  first  and  second 
differences,  which  give,  from  (42), 

2 

jjn         ±j  — j-      u,  j    \     -       _          u/2  V^v 

and 

7ia  —  w  , , 


TIME  OF  MERIDIAN  PASSAGE.  157 

in  which  Ln  is  the  unknown  common  longitude  at  conjunction,  and 
n  in  the  second  member  is  the  required  fractional  portion  of  the 
interval  between  the  consecutive  epochs  of  the  tables. 
Subtracting  and  collecting  the  terms 


(c) 


from  which  n  is  found  by  solution;  the  corresponding  portion  of 
the  constant  tabular  interval  is  then  added  to  T,  thus  giving  the 
Greenwich  time  of  conjunction.  The  time  on  any  meridian  to  the 
west  of  Greenwich  is  found  by  subtracting  the  longitude.  The 
value  of  n  should  be  carried  to  three  places  of  decimals  to  obtain 
the  time  to  the  nearest  minute. 

The  method  of  finding  the  time  of  opposition  is  obvious  from 
the  above,  noting  that  (c)  becomes 

_  d;  +  £t=4   n  =  180°  +  L'  -  L.     (d) 


Except  when  the  moon  is  involved,  the  use  of  first  differences  will 
usually  be  found  sufficient. 

The  times  of  conjunction  and  opposition  in  right  ascension  are 
found  iii  accordance  with  the  same  principles. 

TIME   OF   MERIDIAN   PASSAGE. 

To  determine  the  local  mean  solar  time  of  a  given  body  coming 
to  the  meridian,  it  is  to  be  noted  that  this  time  (P)  is  simply  the 
hour  angle  of  the  mean  sun  at  that  instant,  and  that  this  hour 
angle  is,  by  the  general  formula,  P  =  sidereal  time  —  K.  A.  of  the 
mean  sun. 

Now  the  sidereal  time  at  the  instant  is  equal  to  the  R.  A.  of  the 
body  on  meridian,  and  this  is  equal  to  its  R.  A.  at  the  preceding 
Greenwich  mean  moon  («)  plus  its  increase  of  R.  A.  since  that 
epoch,  which  is  equal  to  m  (P  +  A),  A  being  the  longitude  from 
Greenwich,  and  m  the  body's  hourly  increase  in  R.  A.  Or,  sidereal 
time  =  a  -f-  m  (P  -f-  A). 

Similarly  we  have,  denoting  the  hourly  increase  of  mean  sun's 
R.  A.  by  s,  R.  A.  of  mean  sun  =  aa  +  s  (P  +  ^)- 


158  PRACTICAL  ASTRONOMY. 

Therefore  by  the  preceding  formula, 

P  =  [>  +  m  (P  +  A)]  -  [a.+  s  (P  +  A)]. 

Since  m  and  s  denote  seconds  of  change  per  hour,  A  and  P  in 
the  second  member  are  expressed  in  hours,  and  m  (P  -f  A)  and 
s  (P  -f-  A)  as  also  a  and .  ixs  in  seconds;  therefore  P  in  the  first 
member  is  expressed  in  seconds.  To  express  it  in  hours,  we  have 

[a  +  m  (P  +  A)]  -  [a.  -f  8  (P  +  A)] 
3600 

Solving,  we  have 


3600  —  (m  —  s) 

In  this  equation  a  and  as  are  given  directly  in  the  Ephemeris, 
A  is  supposed  to  be  known,  and  s  is  constant  and  equal  to  9.8565 
seconds;  m  is  obtained  from  the  column  adjacent  to  the  one  giving 
yalue  of  a,  and  should  be  taken  so  that  its  value  will  denote  the 
change  at  the  middle  instant  between  the  Greenwich  mean  moon 
and  the  instant  under  discussion,  viz.,  %  (P  +  A),  as  near  as  can  be 
determined. 

For  the  moon,  whose  motion  in  R.  A.  is  varied,  and  for  an  in- 
ferior planet,  a  second  approximation  may  be  necessary.  If  the 
planet  have  a  retrograde  motion,  m  becomes  negative.  If  the  body 
be  a  star,  m  becomes  zero. 

If  the  sidereal  time  of  culmination  be  required,  the  above 
formula  holds,  substituting  for  the  mean  sun  the  vernal  equinox, 
whose  R.  A.  and  hourly  motion  in  R.  A.  are  zero. 

Hence, 

a  +  Xm 

3600  —  m 
For  a  star,  P'  =  a. 

AZIMUTHS. 

Definitions. — In  surveys  and  geodetic  operations  it  often  becomes 
necessary  to  determine  the  "azimuth"  of  lines  of  the  survey;  i.e., 
the  angle  between  the  vertical  plane  of  the  line  and  the  plane  of  the 
true  meridian  through  one  of  its  extremities;  or,  in  other  words, 
the  true  bearing  of  the  line. 


AZIMUTHS.  169 

For  reasons  given  under  the  head  of  Latitude,  the  geodetic  may 
differ  slightly  from  the  astronomical  azimuth  of  a  line.  Only  the 
latter  will  be  referred  to  here,  and  it  is  manifestly  the  angle  at  the 
astronomical  zenith  included  between  two  vertical  circles,  one  coin- 
ciding with  the  astronomical  meridian,  and  the  plane  of  the  other 
containing  the  line  in  question. 

Outline. — In  outline,  the  method  consists  ins  measuring  with  the 
"  Altazimuth  "  or  "  Astronomical  Theodolite  "  the  horizontal  angle 
which  is  included  between  the  line  and  some  celestial  body  whose 
R.  A.  and  Declination  are  well  known.  Then  having  ascertained 
by  computation  the  true  azimuth  of  the  body  at  the  instant  of  its 
bisection  by  the  vertical  wire,  the  sum  of  the  two  will  be  the  true 
azimuth  of  the  line.  As  will  be  shown  later,  the  celestial  bodies 
best  adapted  for  the  determination  of  azimuths  are  circumpolar 
stars.  For  this  reason  azimuths  in  surveys  and  geodetic  work  are 
usually  reckoned  from  the  North  Point  through  the  East  to  360°. 

Instruments.— The  "  Astronomical  Theodolite  "  is  provided  with 
both  horizontal  and  vertical  circles.  In  geodetic  work  the  latter  is 
used  largely  as  a  mere  finder, 
but  the  former  is  often  of 
great  size — usually  from  one 
to  two  feet  in  diameter,  and 
very  accurately  graduated 
throughout.  For  reading 
the  circle,  it  is  provided 
with  several  reading-micro- 
scopes fitted  with  microm- 
eters, in  lieu  of  verniers; 
and  in  order  that  any  angle 
may  be  measured  with  dif- 
ferent parts  of  the  circle, 
the  latter  is  susceptible  of 
motion  around  the  vertical 
axis  of  the  instrument. 
Eccentricity  and  errors  of 
graduation  are  thus  in  a 
measure  eliminated.  FIO.  31. 

To  mark  the  direction  of  the  line  at  night  a  bull's-eye  lantern 
in  a  small  box  firmly  mounted  on  a  post  is  ordinarily  used ;  the 


160  PRACTICAL  ASTRONOMY. 

light  being  thrown  through  an  aperture  of  such  size  as  to  present 
about  the  same  appearance  as  the  star  observed.  To  avoid  refocus- 
ing  for  the  star,  the  lantern  should  be  distant  not  less  than  a  mile. 
If  it  is  impracticable  to  place  the  lantern  exactly  on  the  line  whose 
azimuth  is  required  it  may  be  placed  at  any  convenient  point,  its 
azimuth  determined  at  night,  and  the  angle  between  it  and  the  line 
measured  by  day;  the  aperture  being  then  covered  symmetrically 
by  a  target  of  any  approved  pattern.  For  convenience  in  the 
following  discussion  the  target  will  be  supposed  to  be  on  the  line. 

Classification  of  Azimuths.  —  Azimuths  of  the  line  with  refer- 
ence to  the  star  are  taken  in  "  sets,"  the  number  of  measurements 
of  the  angle  in  each  set  being  dependent  upon  whether  the  final 
result  is  to  be  a  primary  or  secondary  azimuth.  Primary  azimuths 
are  employed  in  determining  the  direction  of  certain  lines  con- 
nected with  the  fundamental  or  primary  triangulation  of  a  survey, 
and  each  set  consists  of  from  4  to  G  measurements  of  the  angle  in 
each  position  of  the  instrument.  The  final  result  is  required  to 
depend  upon  several  sets,  with  stars  in  different  positions  (generally 
not  less  than  five-,  and  often  many  more).  The  error  of  the  chro- 
nometer (required  in  the  reductions),  together  with  its  rate,  are  de- 
termined by  very  careful  time  observations  with  a  transit. 

Secondary  azimuths  are  employed  in  determining  the  direction 
of  certain  lines  connected  with  the  secondary  or  tertiary  triangles 
of  a  survey.  The  number  of  measurements  in  a  set  is  about  one 
half  or  one  third  that  in  a  set  for  a  primary  azimuth;  the  number 
of  sets  is  also  reduced,  and  the  time  observations  are  usually  made 
with  a  sextant.  The  sun  is  used  in  connection  with  secondary 
azimuths  only. 

Selection  of  Stars.  —  The  true  azimuth  of  the  star  at  the  instant 
of  measuring  the  horizontal  angle  between  it  and  the  line  is  ob- 
tained by  a  solution  of  the  Astronomical  Triangle.  In  order  to 
make  such  a  selection  of  stars  that  errors  in  the  assumed  data  shall 
have  a  minimum  effect  on  the  star's  computed  azimuth,  we  have 

(178) 


.    ^ 

cos  0  tan  d  —  sin  0  cos  P 

Errors  in  the  assumed  values  of  P,  0,  or  d  will  produce  errors  in 
the  computed  azimuth,  those  in  6  being  for  obvious  reasons  usually 
insignificant  and  least  likely  to  occur. 


AZIMUTHS.  161 

Taking  the  reciprocal  of  (178),  differentiating  and  reducing  the 
first  term  of  the  resulting  second  member  by 

cos  a  cos  tp  =  sin  0  cos  d  —  cos  0  sin  d  cos  P9 

the  second  by 

/ 
sin  a  =  sin  0  sin  d  +  cos  0  cos  #  cos  -P> 

and  the  third  by 

sin  J :  sin  P : :  cos  $  :  cos  «, 
we  have 

cos  d  cos  ib  7  „  .              .      .  ,  ,       sin  */>  ,  - 
d  A  = — -d  P  4-  tan  a  sm  ^4  d  0 r  d  d. 

COS  #  COS  # 

From  this  equation  it  is  seen  that  if  we  select  a  close  circumpolar 
star,  any  error  (d  P)  in  the  clock  correction  or  in  the  star's  R.  A., 
or  any  error  (d  0)  in  the  assumed  latitude,  will  produce  but  slight 
effect  on  the  computed  azimuth,  since  cos  d  and  sin  A  will  each  be 
very  small.  If  in  addition  the  star  be  at  elongation  (fi  =  90°),  the 
first  mentioned  error  will  produce  no  effect,  while  sin  A,  although 
at  a  maximum  for  the  star,  will  still  be  very  small.  (In  latitude  of 
West  Point  the  azimuth  of  Polaris  does  not  exceed  1°  40'.)  At 
elongation  the  effect  of  errors  (d  d)  in  d  will  be  a  maximum, 
although  insignificant  if  d  be  taken  from  the  Ephemeris. 

But  if  the  star  be  observed  at  both  east  and  west  elongations, 
the  effect  of  d  d  and  d  0  will  disappear  in  the  mean  result,  since 
the  computed  azimuth  (reckoned  from  the  north  through  the  east  to 
360°),  if  erroneous,  will  be  as  much  too  large  in  one  case  as  too 
small  in  the  other. 

Circumpolar  stars  at  their  elongations  (both)  are  most  favorably 
situated,  therefore,  for  the  determination  of  azimuths;  and  since 
experience  gives  a  decided  preference  to  stars  in  these  positions, 
other  cases  will  not  be  considered,  except  to  remark  that  the  As- 
tronomical Triangle  then  ceases  to  be  right  angled. 

The  stars  a  (Polaris),  d,  and  A,  Ursae  Minoris,  and  51  Cephei, 
are  those  almost  exclusively  used  (although  the  latter  two  cannot 
be  used  with  small  instruments).  Their  places  are  given  in  a 


162  PRACTICAL  ASTRONOMY. 

special  table  of  the  Ephemeris,  pp.  302-13,  for  every  day  in  the 
year,  and  they  are  so  distributed  around  the  pole  that  one  or  more 
will  usually  be  available  for  observation  at  some  convenient  hour. 
Of  these  four,  A  Ursae  Minoris  is  both  the  smallest  and  nearest  to 
the  pole.  For  the  large  instruments  it  therefore  presents  a  finer 
and  steadier  object  than  any  of  the* others.  For  the  small  instru- 
ments suitable  stars  may  be  selected  from  the  Ephemeris. 

Measurements  of  Angles  with  Altazimuth. — In  order  to  under- 
stand the  measurement  of  the  difference  of  azimuth  of  two  points 
at  unequal  altitudes,  let  us  suppose  that  the  horizontal  circle  of  the 
"  Altazimuth  "  has  its  graduations  increasing  to  the  right  (or  like 
those  of  a  watch-face),  and  that  absolute  azimuths  are  reckoned 
from  the  north  point  through  the  east  to  360°,  the  origin  of  the 
graduation  being  at  the  point  0,  Figure  32. 

The  angle  N  L  0  will  then  be  the  absolute  azimuth  of  the  origin 


FIG.  32. 


of  graduations  =  0,  and  if  the  instrument  be  in  adjustment  and 
As  and  AI  denote  the  absolute  azimuths  of  the  star  and  line  respec- 
tively, we  shall  have 


* 


in  which  R  and  R'  denote 


angles  0  L  S  and  0  L  L'  respectively,  and  may  be  considered  as  the 
readings  of  the  instrument  when  pointed  upon  the  star  and  over 
the  line.  These  equations  will  be  somewhat  modified  if  the  instru- 
ment be  not  in  perfect  adjustment.  •  .  This  will  usually  be  the  case. 
Let  us  suppose  that  the  end  of  the  telescope  axis  to  the  observer's 
left  is  elevated  so  that  the  axis  has  an  inclination  of  u  seconds  of  arc. 
Then  if  the  telescope  be  horizontal  and  pointing  ii?  the  direction 


AZIMUTHS.  163 

L  S,  it  will,  when  moved  in  altitude,  sweep  to  the  right  of  the  star, 
and  the  whole  instrument  must  be  moved  to  the  left  to  bring  the 
line  of  collimation  on  the  star.  The  reading  of  the  instrument  will 
thus  be  diminished  to  r,  and  we  shall  have  the  proper  reading, 
R  —  r  -\-  a  correction.  The  amount  of  this  correction  is  readily 
seen,  from  the  small  right-angled  spherical  triangle  involved  (of 
which  the  required  distance  is  the  base),  to  be  I  cot  z.  In  the  same 
way  it  is  seen  from  the  principles  explained  under  "  Equatorial 
Intervals,"  etc.,  that  if  the  middle  wire  be  to  the  left  of  the  line  of 
collimation  by  c  seconds  of  arc,  r  must  receive  the  correction 
c  cosec  z.  Hence  when  both  these  errors  exist  together,  we  shall 
have,  z'  denoting  the  zenith  distance  of  the  target, 


A8  =  0  -f-  r  -f-  1  cot  z  +  c  cosec  z, 

Al  —  0  -f  r'  +  V  cot  z9  -f  c  cosec  z',  (180) 

since  c  remains  unchanged,  while  b  is  subject  to  changes. 
Subtracting, 

AI  —  Aa  =  (r'  +  V  cot  z')  —  (r  -{-  I  cot  z)  4-  c  (cosec  z'  —  cosec  z). 

Since  by  reversing  the  instrument  the  sign  of  c  is  changed,  but 
not  altered  numerically,  we  may,  if  an  equal  number  of  readings  in 
the  two  positions  be  taken,  drop  the  last  term  as  being  eliminated 
in  the  mean  result.  With  this  understanding,  the  equation  will  be 

Al  -  A8  =  (r'  +  V  cot  z')  -(r+b  cot  z).  (181) 

which  gives  the  azimuth  of  the  line  with  reference  to  the  star, 
free  from  all  instrumental  errors,  b  is  positive  when  the  left 
end  is  higher,  and  its  value,  heretofore  explained,  is  obtained  by 
direct  and  reversed  readings  of  both  ends  of  the  bubble,  and  is 

-     (w  -f  w')  —  (e  +  e')   \,d  being  the  value  of  one  division  in 

seconds  of  arc.  For  stars  at,  or  very  near,  elongation,  it  is  evident 
that  cot  z  may  be  replaced  by  tan  0,  without  material  error;  c  is 
positive  when  middle  wire  is  to  the  left  of  its  proper  position. 

For  very  precise  work  the  above  result  requires  a  small  correc- 
tion for  diurnal  aberration,  the  effect  of  which  is  to  displace  (appar- 


164  PRACTICAL  ASTRONOMY. 

ently)  a  star  toward  the  east  point.  For  stars  at  elongation,  this 
correction  is  0".311  cos  Ae.  (See  Note  1.) 

In  using  the  reading-microscopes,  care  should  be  taken  to  correct 
for  "  error  of  runs."  When  a  microscope  is  in  perfect  adjustment, 
a  whole  number  of  turns  of  the  micrometer  screw  carries  the  wire 
exactly  over  the  space  between  two  Consecutive  graduations  of  the 
circle.  Due  to  changes  of  temperature,  etc.,  the  distance  between 
the  micrometer  and  circle  may  change,  thus  altering  the  size  of  the 
image  of  a  "  space."  The  excess  of  a  circle  division  over  a  whole 
number  of  turns  is  called  the  "  Error  of  Runs."  This  error  is  de- 
termined by  trial,  and  a  proportional  part  applied  to  all  readings  of 
minutes  and  seconds  made  with  the  microscope. 

Observations  and  Preliminary  Computations. — The  observations 
and  the  preliminary  computations  are  as  follows:  The  error  and 
rate  of  the  chronometer,  error  of  runs  of  the  micrometers,  collima- 
tion  error  and  latitude  are  supposed  to  have  been  obtained  with 
considerable  accuracy.  The  apparent  R.  A.  and  declination  for  the 
time  of  elongation  of  the  star  to  be  used  must  be  taken  from  the 
Ephemeris,  or,  if  "not  given  there,  reduced  from  the  mean  places 
given  in  the  catalogue  employed,  as  explained  under  Zenith 
Telescope. 

Then  for  the  star's  hour-angle  at  elongation,  cos  Pe  =  -  -~. 

belli  O 

OL 

«      azimuth      "  «          sin  Ae  =  —  -.. 

COS  0 

"          "         "      zenith  distance  at    "  cos  ze  =  —. — -r. 

sm  d 

"          "    sidereal          time  "    "  T0  =  a  ±  Pe. 

"    chronometer    "  "    "  Tc  =  T0  —  E, 

a  being  the  R.  A.,  and  E  the  chronometer  error. 

The  instrument  is  then  placed  accurately  over  the  station  and 
levelled,  so  that  everything  will  be  in  readiness  to  begin  observations 
at  about  20m  before  the  time  of  elongation  as  above  computed.  In 
the  actual  measurement  of  the  angle  several  different  methods  have 
been  followed.  First,  five  or  six  pointings  are  made  on  the  target, 
and  for  each  pointing,  the  circle  and  all  the  microscopes  are  read; 
also  if  the  angle  of  elevation  of  the  target  differ  sensibly  from  zero 
(as  would  not  usually  be  the  case  with  the  base-line  of  a  survey) 
readings  of  the  level,  both  direct  and  reversed,  are  made.  If  the 


AZIMUTHS.  165 

target  be  on  the  same  level  as  the  instrument,  cot  z'  will  be  zero, 
and  the  level  correction  will  disappear.  Then  five  or  six  pointings 
are  made  on  the  star,  and  in  addition  to  the  above  readings  the 
chronometer  time  of  each  bisection  is  noted.  The  instrument  is 
then  reversed  to  eliminate  error  of  collimation,  and  the  above 
operations  repeated,  beginning  with  the  star.  In  the  second  method 
alternate  readings  are  made  on  the  mark  and  star,  star  and  mark, 
until  five  or  six  measurements  of  the  angle  have  been  made,  the 
chronometer  being  read  at  each  bisection  of  the  star;  the  circle, 
microscopes  and  level  as  before.  The  instrument  is  then  reversed, 
and  the  same  operations  repeated  in  the  reverse  order.  The  middle 
of  the  time  occupied  by  the  whole  set  should  correspond  very  nearly 
to  the  time  of  elongation.  Similar  observations  are  then  made,  on 
the  same  or  following  nights,  on  other  stars,  combining  both  eastern 
and  western  elongations,  and  using  different  parts  of  the  horizontal 
circle  for  the  measurement. 

Reduction  of  Observations.— Since  the  observations  on  the  star 
have  been  made  at  different  times,  and  since  these  correspond  to 
different  though  nearly  equal  azimuths,  the  first  step  in  the  reduc- 
tion is  to  ascertain  what  each  reading  on  the  star  would  have  been 
had  the  observation  been  made  exactly  at  elongation.  For  this- 
purpose  find  the  difference  between  the  chronometer  time  of  each 
observation  and  the  chronometer  time  of  elongation  as  computed, 
applying  the  rate  if  perceptible.  Let  the  sidereal  interval  between 
these  two  epochs  be  denoted  by  r  seconds.  Then  the  elongation 
reading  of  the  star  would  have  been 

actual  reading  ±  the  expression  112.5  r2  sin  1"  tan  Ae , 

which  denote  by  C.     (See  Note  2.) 

[The  quantity  112.5  r2  sin  1"  is  almost  exactly  equal  to  the 
tabulated  values  of  "  m  "  in  the  "  Reduction  to  the  Meridian,"  and 
may  if  desired  be  taken  directly  from  those  tables.]  With  a 
circle  graduated  as  assumed,  this  correction  would  manifestly  be 
negative  for  a  western,  and  positive  for  an  eastern,  elongation. 
Hence  Eq.  (181)  becomes, 

Al  -  Ae  =  (r'  +  b'  cot  *')  -  (r  +  b  cot  z  ±  C).          (182) 
Each  pair  of  observations  (on  the  line  and  star)  with  the  telescope 


166  PRACTICAL  ASTRONOMY. 

"direct  "  gives  a  value  of  Al  —  Ae.     If  nd  be  the  number  of  such 
pairs,  the  mean  will  be  —  —      —  ~,  to  which  if  Ae  (positive  for 

eastern,  negative  for  western,  elongations)  be  added  as  heretofore 

(cos  6  \ 
sin  Ae  —  --  37  I,  we  shall  have  the  true  bearing  of  the 
cos  0; 

line  for  instrument  "  direct" 

Similarly,  for  instrument  "  reversed,"  we  shall  have 

^(  A,  -  Ae) 


from  which  by  adding  ^4ewe  obtain  the  true  bearing  of  the  line  for 
instrument  reversed. 

The  mean  of  the  two  is  the  true  hearing  of  the  line  as  given  by 
the  star  employed. 

[For  the  greatest  precision,  this  must  be  corrected  by  adding 
the  diurnal  aberration,  0".311  cos  Ae.] 

The  adopted  value  of  the  azimuth  of  the  line  should  rest  upon 
<ai  least  five  such  determinations. 


E 
FIG.  33. 


»|«  1.  Diurnal  Aberration  in  Azimuth. 

It  has  already  been  shown  when  treating  of  the  transit  instru- 
ment and  in  Art.  225  Young,  that  due  to  diurnal  aberration  all 
stars  are  apparently  displaced  toward  the  east  point  of  the  horizon 
by  0".319  cos  0sin  Q  of  a  great  circle;  where  0  is  the  angle  made  by 
the  direction  of  a  ray  of  light  from  the  star  with  an  east  and  west 
line  (measured  by  8  E,  Fig.  33). 

To  determine  the  effect  of  this  small   displacement   on   the 


AZIMUTHS.  167 

azimuth  of  a  star,  the  right-angled  triangle  Z  8  M  gives,  denoting 
ZMby  I, 

cos  6 

sin  A  =  — 

sin  2 

sin  z  cos  ^4  =  sin  6  sin  5. 
Hence 

cos  0  cos  J. 


sin  A 


sin  6  sin  b  ' 


.     cote 

tan  A  =  — — 7. 
sin  o 

Differentiating 


. 

sin  &  sin  o'  sin  0  sin  z 

Substituting  —  0".319  cos  0  sin  0  f or  d  0  (since  0  is  a  decreasing 
function  of  ^4), 

0".319  cos  ^  cos  0 
d  A  —  — - —   — -. • 

sin  z 

For  a  close  circumpolar  star  at  elongation 

cos  0  =  sin  z,    sensibly. 
Hence, 

d.4  =  0".319cos  Ae.  (183) 

»J*  2.  To  Reduce  an  Azimuth  Observed  Shortly  Before  or  After 
the  Time  of  Elongation,  to  its  Value  at  Elongation. 

If  we  conceive  the  meridian  to  be  revolved  to  the  position  of  the 
declination  circle  passing  through  the  point  of  elongation,  evidently 
the  arc  of  this  circle  intercepted  between  the  vertical  wire  of  the 
instrument  and  the  point  of  elongation  will  have  the  same  numeri- 
cal value  as  the  "  Eeduction  to  the  Meridian"  deduced  in  connec- 
tion with  the  Zenith  Telescope,  viz. : 

J  (15r)2  sin  1"  sin  2  d  =  112.5  rz  sin  1"  sin  tf  cos  d. 


168  PRACTICAL  ASTRONOMY. 

The  angle  at  the  zenith  subtended  by  this  arc,  i.e.,  the  correction 
to  azimuth,  is  seen  from  the  small  right-angled  triangle  to  be 


(184) 


smze 


Substituting  cos  d  and  sin  d  for  sin  6  and  cos  d  (d  =  polar  dis- 
tance), making  cos  p  —  1,  and  sin  p  =  tan^>  (since  the  star  is  a  close 
circumpolar),  the  last  factor  becomes 

*?JJ?  =  tan  Ae.    Hence  O=  112.5  r*  sin  1"  tan  A* 


DECLINATION  OF  THE  MAGNETIC  NEEDLE. 

The  Declination  of  the  Magnetic  Needle  may  be  found  in  ac- 
cordance with  the  same  principles,  regarding  the  magnetic  meridian 
pointed  out  by  the  needle,  as  the  line  whose  azimuth  is  to  be  found. 
Or,  note  the  reading  of  the  needle  when  the  instrument  carrying  it 
is  pointed  accurately  along  a  line  whose  true  bearing  or  azimuth  is 
known.  Or,  take  the  magnetic  bearing  of  some  known  celestial 
body,  and  note  the  time  T.  Then  P  =  T  -  a.  This  value  of  P 
in  Eq.  (178)  gives  the  true  azimuth,  and  the  difference  between 
this  and  the  magnetic  bearing  gives  the  declination  of  the  needle. 
Or,  if  the  time  be  not  known,  measure  the  altitude  of  the  body  and 
solve  the  Z  P  S  triangle  for  A,  knowing  0,  #,  and  a.  Then  having 
noted  the  magnetic  bearing  of  the  body  at  the  instant  of  measuring 
the  altitude,  the  difference  is  the  decimation  of  the  needle. 

One  of  the  most  accurate  methods  of  laying  out  the  true  merid- 
ian is  by  means  of  a  Transit  Instrument  adjusted  to  the  meridian, 
and  whose  instrumental  errors  a  and  c  have  been  carefully  deter- 
mined by  star  observations. 

SUN-DIALS. 

A  sun-dial  is  a  contrivance  for  indicating  apparent  solar  time 
by  means  of  the  shadow  of  a  wire  or  straight-edge  cast  on  a  properly 
graduated  surface.  The  wire  or  straight-edge,  called  the  style  or 
gnomon,  must  be  parallel  to  the  earth's  axis;  i.e.,  it  must  be  inclined 
to  the  horizontal  by  an  angle  equal  to  the  latitude,  and  be  in  the 


SUN-DIALS. 


169 


meridian.  The  graduated  surface,  called  the  dial-face,  is  usually 
a  plane,  and  made  either  of  metal  or  smoothed  stone.  It  may  have 
any  position  with  reference  to  the  style  (consistent  with  receiving 
its  shadow  throughout  the  day),  although  it  is  usually  either  hori- 
zontal or  placed  in  the  prime  vertical.  The  two  varieties  are  shown 
in  Fig*  a,  the  first  being  by  far  the  more  common. 


FIG.  a. 


The  principle  of  the  horizontal  dial  will  be  readily  understood 
from  an  inspection  of  Fig.  b. 

Let  PP9  be  the  axis  of  the  celestial  sphere,  ^the  zenith,  A  Q  B 
the  equinoctial,  and  A  H  B  perpendicular  to  OZihe  plane  of  the 
dial  face,  the  style  extending  from  0  in  the  direction  of  P.  Then 
if  a  plane  be  passed  through  the  style  and  the  position  of  the  sun, 
S,  at  any  instant,  it  will  cut  from  the  celestial  sphere  the  sun's 
hour-circle,  and  from  the  dial-face  the  line  G IX,  which  is  therefore 
the  shadow  of  the  style  on  the  dial-face.  The  direction  of  this  line 
is  thus  seen  to  be  independent  of  the  sun's  declination  (season  of 
the  year),  and  dependent  only  on  his  hour  angle.  If ,  therefore,  we 
mark  on  the  dial-face  the  various  positions  of  this  line  correspond- 
ing to  assumed  hour  angles  which  differ  from  each  other  by,  for 


170 


PRACTICAL  ASTRONOMY. 


example,  3°  45'  or  15  minutes,  instants  of  apparent  solar  time  will 
be  indicated  by  the  arrival  of  the  style's  shadow  at  the  correspond- 
ing line.  This  construction  may  be  made  as  follows,  noting  that 


FIG.  6. 

the  12-o'clock  line  is  the  intersection  of  the  dial-face  with  the 
vertical  plane  through  the  style. 

Suppose,  for  example,  it  were  required  to  construct  the  9-o'clock 
line.  In  the  spherical  triangle  P  H'  IX  right-angled  at  H'  we 
have  P  H'  =  </>,  and  the  angle  at  P  =  Z  P  S  =  45°,  to  determine 
the  side  H'  IX '=  x,  given  by  the  formula 

tan  x  =  sin  0  tan  45°. 

Then  with  O  as  a  center  lay  off  an  angle  from  CH'  equal  to  the 
computed  value  of  x,  and  draw  the  line  O IX. 
Generally, 

tan  x  =  sin  0  tan  P9 

P  denoting  the  hour  angle  assumed. 

Values  of  x  corresponding  to  intermediate  values  of  P  may  be 
laid  off  with  a  pair  of  dividers. 

The  dial-face  may  have  any  convenient  form, — circular,  rectan- 
gular, or  elliptical.  The  last  is  the  best  form  (shown  in  Fig.  «), 
since  the  axis  can  be  so  proportioned  that  the  spaces  along  the  edge 


SUN-DIALS.  171 

will  be  nearly  equal,  thus  greatly  facilitating  any  subdivision.  For 
the  latitude  of  West  Point,  C  IF  should  be  about  2£  times  C  A 
(Fig.  a}.  If  the  plate  be  18  or  20  inches  long  the  subdivisions  can 
be  readily  carried  to  minutes. 

Usually  the  style  is  a  triangular-shaped  piece  of  metal  of  a  suf- 
ficient thickness  to  avoid  deformation  by  accident — say  i  or  J  inch. 
In  this  case  one  edge  will  cast  the  shadow  in  the  A.M.,  and  the 
other  in  the  P.M.  Hence  the  graduations  on  either  side  of  the 
12-o'clock  line  must  be  constructed  using  as  a  center  the  point 
where  the  shadow-casting  edge  pierces  the  plane  of  the  dial-face. 
The  plane  of  the  style  must  be  accurately  perpendicular  to  the 
dial-face. 

Having  been  graduated,  the  sun-dial  is  mounted  on  a  firm 
pedestal,  accurately  levelled  by  a  spirit-level,  and  turned  till  the 
plane  of  the  style  is  in  the  meridian.  For  an  approximation  we 
may  use  a  pocket  compass,  the  declination  of  the  needle  being 
known  within  moderate  limits.  By  day  the  orientation  may  be 
effected  by  means  of  a  watch  whose  error  is  known.  Compute  the 
watch  time  of  apparent  noon  =  12-o'clock  —  error  -f  equation  of 
time,  and  turn  the  dial  slowly,  keeping  the  shadow  of  style  on  the 
12-o'clock  mark  until  the  time  computed.  The  levelling  must  be 
carefully  attended  to.  If  the  watch  error  be  not  known,  it  may  be 
.found  by  means  of  a  sextant. 

If  no  means  of  determining  time  are  at  hand,  the  dial  may  still 
be  oriented  by  a  determination  of  the  meridian  plane,  either  by 
day  or  night.  At  night  advantage  may  be  taken  of  the  fact  that 
Polaris  and  C  Ursae  Majoris  (the  middle  star  in  the  tail  of  the 
Great  Bear  or  handle  of  the  Dipper)  cross  the  meridian  at  almost 
exactly  the  same  instant.  Therefore  if  two  plumb-lines  be  sus- 
pended from  firm  supports  as  nearly  in  the  meridian  as  may  be,  one 
touching  the  style  and  the  other  a  few  feet  to  the  south  arranged 
for  lateral  shifting,  we  may  by  sliding  the  latter  cover  both  stars  by 
both  lines  at  the  moment  of  meridian  passage.  These  lines  then 
define  the  meridian  plane,  into  which  the  style  is  easily  turned. 
The  polar  distance  of  C  being  between  34°  and  35°,  it  is  evident 
that  for  latitudes  above  about  40°  the  star  must  be  observed  at 
lower  culmination,  and  for  lower  latitudes  at  the  upper. 

By  day  the  meridian  plane  may  be  determined  as  follows:  Sus- 
pend a  plumb-line  over  the  south  end  of  a  perfectly  level  table  or 


172 


PRACTICAL  ASTRONOMY. 


other  suitable  surface.  With  the  point  A  as  a  center  describe  an 
arc,  CD.  The  shadow  of  a  knot  or  bead  at  B  will  describe  during 
the  day  a  curve  G  E  F  G  D.  Mark  the  points  C  and  D  where  it 
crosses  the  arc  before  and  after  noon.  A  line  from  A  bisecting  the 
chord  G  D  will  then  be  in  the  meridian,,  and  its  extremities  may  be 
projected  to  the  earth  by  pluntfe-lines  and  the  points  marked. 
Stretch  a  fine  cord  from_one  point  to  the  other,  and  note  the  instant 


FIG.  c. 

when  the  shadow  of  the  south  plumb-line  exactly  coincides  with 
that  of  the  cord.  This  is  evidently  apparent  noon;  and  if  the  dial 
be  so  turned  that  the  shadow  of  the  style  falls  on  the  12 -o'clock 
line  at  the  same  instant,  it  will  be  duly  oriented. 

Evidently  this  method  supposes  the  sun's  declination  to  be  con- 
stant; its  change  may,  however,  for  this  purpose  be  neglected, 
except  for  a  month  at  about  the  time  of  the  equinoxes. 

The  meridian  line  may  also  be  determined  with  a  theodolite,  as 
described  in  works  on  Surveying. 

The  dial-face  may  if  desired  be  graduated  after  orientation  by 
noting  where  the  shadow  of  the  style  falls  at  1  hour,  2  hours,  etc., 
from  the  time  of  apparent  noon. 

The  indications  of  all  sun-dials  must  be  corrected  by  the  Equa- 
tion of  Time  in  order  to  give  local  mean  time.  This  correction  is 
practically  constant  for  the  corresponding  days  of  all  years,  and  its 
value  at  suitable  intervals  may  either  be  engraved  on  the  dial-plate, 
or  taken  from  the  annexed  table. 

Refraction,  varying  witji  the  sun's  altitude,  is  evidently  a  source. 


SOLAR  ECLIPSE. 


173 


of  error,  although  too  small  to  require  consideration  in  the  present 
connection. 

The  indications  of  a  sun-dial  with  the  solid  style  (Fig.  a)  will 
be  one  minute  too  great  in  the  forenoon  and  one  minute  too  small 
in  the  afternoon,  since  the  shadow  line  will  in  each  case  be  formed 
by  the  limb  of  the  sun  toward  the  meridian,  and  the  sun  requires 
about  one  minute  to  advance  through  an  arc  equal  to  its  semi- 
diameter. 

A  dial  constructed  for  a  given  latitude  may  be  used  without 
appreciable  error  in  any  latitude  not  differing  therefrom  by  more 
than  one  third  of  a  degree — say  25  miles. 

Vertical  dials  are  usually  placed  on  the  south  fronts  of  buildings. 
Their  construction  is  readily  understood  from  what  precedes,  the 
graduations  being  computed  by  the  formula 

tan  x  =  cos  0  tan  P. 
EQUATION  OF  TIME— TO  BE  ADDED  TO  SUN-DIAL  TIME. 


Day. 

Jan. 

Feb. 

March. 

April. 

May. 

June. 

1 

+  4™ 

4-14™ 

+  12™ 

+  4™ 

_3m 

-2™ 

8 

7 

14 

11 

2 

-4 

-1 

16 

10 

14 

9 

0 

-4 

0 

24 

12 

13 

6 

-2 

-3 

+  2 

Day. 

July. 

Aug. 

Sept. 

Oct. 

Nov. 

Dec. 

1 

+  3- 

-f  6™ 

Qm 

-10™ 

-IQm 

-10" 

8 

5 

5 

-2 

-12 

-16 

-  7 

16 

6 

4 

-5 

-14 

-15 

-  4 

24 

6 

2 

-8 

-15 

-13 

0 

SOLAR  ECLIPSE.  ;> 

A  solar  eclipse /can  only  occur  at  conjunction — that  is,  at  new 
moon,  and  then  only  when  the  moon  is  near  enough  to  the  plane  of 
the  ecliptic  to  throw  its  shadow  or  penumbra  upon  the  earth.  The 
following  discussion,  abbreviated  from  that  found  in  Chauvenet's 
Practical  Astronomy,  Vol.  I,  will  suffice  to  give  the  student  such  a 


174  PRACTICAL  ASTRONOMY. 

knowledge  of  the  theory  of  eclipses  as  to  enable  him  to  project 
a  solar  eclipse,  with  the  aid  of  the  eclipse  data  found  in  the 
Ephemeris. 

Solar  Ecliptic  Limits.— Let  N  S  Fig.  34  be  the  Ecliptic,  N  M 


s  s1      P 

FIG.  34. 


the  intersection  of  the  plane  of  the  moon's  orbit  with  the  celestial 

sphere,  JV^the  moon's  node,  8  and  M  the  sun's  and  moon's  center 

at  conjunction,  and  8'  and  M'  the  same  points  at  the  instant  of 

nearest  angular  distance  of  the  moon  from  the  sun.     Assume  the 

following  notation,  viz.  :  — 

/?  =  8  M9  the  moon's  latitude  at  conjunction. 

i   =  8  N  M9  the  inclination  of  the  moon's  orbit  to  the  ecliptic. 

,\  =  the  quotient  of  the  moon's  mean  hourly  motion  in  longitude  at 

conjunction,  divided  by  that  of  the  sun. 
A  ••=  S'  Mf,  the  least  true  distance. 
y  =SMS'. 

Considering  JV  M  S  as  a  plane  triangle,  and  drawing  the  perpen- 
dicular M'  P  from  M'  to  8  N,  we  have 


$8'  =  /?  tan  ;/.        SP  =  hp  tan  y. 
8'  P  =  ft  (A,  -  1)  tan  y.        M'  P  =  ft  -  \  ft  tan  y  tan  i. 
4*  =  P  [  (1  -  l)a  tan8  y  +  (1  -  A  tan  i  tan  ;/)a  ]. 

Differentiating  the  last  equation  and  placing  —r—  =  0,  we  find  A 
will  be  a  minimum  for 

A  tan  t 


-  I)8  +  A8  tana 
This  value  gives 

-i)' 


SOLAR  ECLIPSE.  '  175 

or 

Ja  =  p  cos2  1'',  (186) 

when  tan  if  is  placed  equal  to  .  _     tan  i. 

The  least  apparent  distance  of  the  sun's  and  moon's  center  as 
viewed  from  the  surface  of  the  earth  may  be  less  than  A  by  the 
difference  of  the  horizontal  parallaxes  of  the  two  bodies.  Call  this 
distance  A'  9  then 

A'  =A—  (n  -  P). 

Now  when  A'  is  less  than  the  sum  of  the  apparent  semi-diameters 
of  the  sun  and  moon  there  will  be  an  eclipse  ;  hence  the  condition 
is  (denoting  the  semi-diameters  of  the  moon  and  sun  respectively 
by  s'  and  s), 


or 

/3  cos  i'  <  TT  -  P  +  s  +  s'.  (187) 

To  ascertain  the  probability  of  an  eclipse,  it  is  generally  suffi- 
cient to  substitute  the  mean  values  of  the  quantities  in  the  above 
inequality.  The  extreme  values,  determined  by  observation  are 


.   -(  5°  20'  06" 
*    |  4°  57'  22" 

(  61'  32" 
(  52'  50" 

p    j  9".  0 

(  8".70 
8".85 

l    j  16.19 
"    (  10.89 

1.00472 

5°    8'  44" 

j  16'  18" 
|  15'  45" 

16'    1" 

i  value  of  sec  i'y 

57'  11" 

,,  j  16'  46" 
\  14'  24" 
15'  35" 

found  from  those 

13.  5 

of  i  and  A,  is 

and  hence, 

(188) 
(it  -  P  +  s  -f-  5')  sec  i'  =  ft  <(TT  -  P  +  s  +  s')  (1  +  0.00472). 


The  fractional  part  of  the  second  member  of  the  inequality 
varies  between  20"  and  30";  taking  its  mean  25",  we  have  for  all 
but  exceptional  cases, 

/?<7r-P  +  s4-s'  +  25".  (189) 


176  PRACTICAL 

Substituting  in  this  last  form  the  greatest  values  of  TT,  $,  and 
s',  and  the  least  value  of  P;  and  then  the  least  values  of  TT,  st  and 
s'y  and  the  greatest  value  of  P,  we  have 

ft  <  lc  34'  27".3, 
and 

ft  <  1°  22'  50", 

respectively. 

If,  therefore,  the  moon's  latitude  at  conjunction  be  greater  than 
1°  34'  27".3  a  solar  eclipse  is  impossible;  if  less  than  1°  22'  50"  it 
is  certain;  if  between  these  values  it  is  doubtful.  To  ascertain 
whether  there  will  be  one  or  not  in  the  latter  case,  substitute  the 
actual  values  of  P,  TT,  s  and  s'  for  the  date,  and  if  the  inequality 
subsists  there  will  be  an  eclipse,  otherwise  not.  . 


PROJECTION    OF    A    SOLAR    ECLIPSE. 

1.  To  find  the  Radius  of  the  Shadow,  on  any  Plane  perpendicular 
to  the  Axis  of  the  Shadow. 

In  Fig.  35  let  S  and  M be  the  centers  of  the  sun  and  moon;  V 
the  vertex  of  the  umbral  or  penumbral  cone;  F  Et\^ fundamental 
plane  through  the  earth's  center  perpendicular  to  the  axis  of  the 
shadow ;  and  C  D  the  parallel  plane  through  the  observer's  position. 
It  is  required  to  find  the  value  of  C  D  at  the  beginning  or  ending 
of  an  eclipse. 

Take  the  earth's  mean  distance  from  the  sun  to  be  unity,  and 

let  ES  =  r,  EM  =  r',  MS  =  r-  r'.     Place  H^L  =  ^,  and  let  k 

be  the  ratio  of  the  earth's  equatorial  radius  to  the  moon's  radius 
=  0.27227.  Then  P0  being  the  sun's  mean  horizontal  parallax,  we 
have 

Earth's  radius  =  sin  P0. 

Moon's  radius  =  k  sin  P0  =  0.27227  sin  P# 

Sun's     radius  =  sin  s. 


SOLAR 


177 


5  being  the  apparent  semi-diameter  of  the  sun  at  mean  distance 
From  the  figure  we  have 


sn 


sin  s  ±  k  sin  P. 


E   F 


FIG.  35. 


(190) 


in  which  the  upper  sign  corresponds  to  the  penumbral  and  the 
lower  to  the  umbral  cone.  The  numerator  of  the  second  member 
is  constant,  and  since  s  =  959".758,  PQ  =  8".85,  we  have 

log  [sin  s  -j-  k  sin  P0]  =  7.6688033  for  exterior  contact, 
log  [sin  s  —  ~k  sin  P0]  =  7.6666913  for  interior  contact. 
If  the  equatorial  radius  of  the  earth  be  taken  as  unity,  we  have 

k 

'  sin/ 


178 


PRACTICAL  ASTRONOMY. 


Whence  the  distance  c  of  the  vertex  of  the  cone  from  the  fun- 
damental plane  is 


If  I  and  L  be  radius  of  the  shadow  on  the  fundamental  and  on  the 
observer's  plane  respectively,  andTc  be  their  distance  apart,  we  have 

I  =  c  tan/  =2  tan  /±  k  sec/.  (192) 

L  =  (c  -  C)  tan  /  =  I  -  C  tan/.  (193) 

2.  To  find  the  distance  of  the  Observer  at  a  given  time  from  the 
Axis  of  the  Shadow  in  terms  of  his  Co-ordinates  and  those  of 
the  Moon's  Center,  referred  to  the  Earth's  Center  as  an  Origin. 

Let  0,  Figure  36,  be  the  earth's  center,  and  X  Y  the  funda- 
mental plane.    Take  Z  Y  to  be  the  plane  of  the  declination  circle 


FIG.  36. 


passing  through  the  point  Z  in  which  the  axis  of  the  moon's  shadow 
pierces  the  celestial  sphere ;  X  Z  being  perpendicular  to  the  other 
two  coordinate  planes.  Let  M  and-#  be  the  centers  of  the  moon 
and  sun,  M' ,  S',  their  geocentric  places  on  the  celestial  sphere,  Mt 
their  projections  on  the  fundamental  plane,  and  (7,  the  projection 
of  the  observer's  place  on  the  same  plane.  Let  P  be  the  north 


SOLAR  ECLIPSE.  179 

pole.  The  axis  0  Z,  being  always  parallel  to  the  axis  of  the  shadow, 
will  pierce  the  celestial  sphere  in  the  same  point,  as  8  M.  Assume 
the  following  notation  : 

a,   #,  r  =  the  R.  A.,  Dec.,  and  distance  from  the  earth's  center, 

respectively,  of  the  moon's  center. 

a',  #',  r'  =  the  corresponding  coordinates  of  the  sun's  center. 
a,  d,          —  the  R.  A.  and  Dec.  of  the  point  Z. 
x,  y,  z       =  the  coordinates  of  the  moon's  center. 
£,  77,  C      =  the  coordinates  of  the  observer's  position. 
0,  0'        =  the  latitude  and  reduced  latitude  respectively. 
A  =  the   longitude  of  the   observer's    station    west    from 

Greenwich. 
p  —  the  earth's  radius  at  the  observer's  station  in  terms  of 

the  earth's  equatorial  radius  taken  as  unity. 
jjLt  =  the  Greenwich  hour  angle  of  the  point  Z. 

IJL  =  the  sidereal  time  at  which  the  point  Z  has  the  R.  A.  a. 

A  —  the  required  distance  of  the  place  of  observation  from 

the  axis  of  the  shadow  at  the  time  yw. 
From  the  conditions,  we  have 

R.  A.  of  Z  =  a, 
R.  A.  of  M'-  a, 

R.  A.  of  X  =  90°  +  a, 
and  therefore 

ZP  M  '  =  a  -  a,    and  P  M'  =  90°  -  tf. 

Through  Mt  and  Gt  draw  Mt  N  and  C4  N  parallel  to  the  axis  of 
Xand  Irrespectively;  then  Mt  C,  N  =  P  Z  M'  =  P,  the  position 
angle  of  the  point  of  contact,  and  we  have 


(194) 


180  PRACTICAL  ASTRONOMY. 

From  the  spherical  triangles  M'  P  X,  Mf  P  Y,  and  Mf  P  Z,  we 
have 

x  —  r  cos  Mf  X  —  r  cos  d  sin  (a  —  a)  } 

y  —  r  cos  M'  Y  =  r  [sin  d  cos  d  —  cos  d  sin  d  cos  (a  —  #)]   >    (1  95) 

z  =  r  cos  M'  Z  —  r  [sin  $  sin  d  -j-  cos  d  cos  6?  cos  (a  —  a)].    J 

Similarly  the  coordinates  of  the  place  of  observation  are 

g  —  p  cos  0'  sin  (//  —  a)  1 

TI  —  p  [sin  0'  cos  d  —  cos  <p'  sin  a?  cos  (^  —  «)]  >•  (LOG) 

C  =  p  [sin  0'  sin  d  -J--  cos  0'  cos  d  cos  (/*  —  «)].  ) 

The  hour  angle  (/*  —  ft)  of  the  point  Z  for  the  meridian  of  the 
observer  can  be  found  from 


in  which  ju,  is  the  hour  angle  of  the  point  Z  for  the  Greenwich 
meridian  and  A  is  the  longitude  of  the  observer's  meridian. 

The  distance  of  the  observer  from  the  axis  of  the  moon's  shadow 
A,  —  Ct  Mt  can  be  found  from  the  above  formulas, 

« 

since,  A*  =  (x  -  £)'  +  (y  -  ?/)\  (197) 

3.     To  Find  the  Time  of  Beginning  or  Ending  of  the  Eclipse  at  the 

Place  of  Observation. 

For  the  assumed  Greenwich  mean  time  of  computation  take 
from  the  Besselian  table  of  elements  given  in  the  Ephemeris  for 
each  eclipse  the  values  of  sin  d,  cos  d,  and  jar  The  values  of  p  cos  0' 
p  sin  0'  are  found  on  page  505,  computed  from  the  formulas, 


,  .  Ct  COS  0 

pcos0  =    —==      =  F  cos  0 

'2J 


(198) 


,       sin  0  V\.  —  e2  sin2  0       sin  0 
-  "  ~~ 


SOLAR  ECLIPSE.  181 

The  variations  of  £  and  //  in  one  minute  of  mean  time  are  ob- 
tained by  differentiating  the  first  two  of  Eqs.  (196),  and  give 

£'  =  [7.63992]  p  cos  0'  cos  (/i,  -  A)  ] 

rf  —  [7.63992]  p  cos  0'  sin  d  sin  (^  —  A)         \  (199) 
=  [7.63992]  £  sin  rf. 

The  variations  of  x  and  y  for  one  minute  of  mean  time  are 
represented  by  x',  and  y',  and  their  logarithms  are  given  in  the 
lower  table  of  the  Ephemeris  elements  for  the  eclipse.  Now,  if  the 
time  chosen  for  computation  be  exactly  the  instant  of  beginning  or 
ending  of  the  eclipse,  then  A  =  L\  but  as  this  is  scarcely  possible 
a  correction  r  in  minutes  must  be  made  to  the  assumed  Ephemeris 
time  T. 

We  may  then  write, 

L  sin  P  =  x  -  Z  +  (x'  -  £')  r,  (200) 

L  cos  P  =  #-//+  (/  -  ?/')  r.  (201) 

Assume  the  auxiliary  quantities  m,  M,  n,  N,  given  by  the  equa- 
tions, 

m  sin  M  =  x  —  & 


(202) 

n  cos  N  —  y'  —  ?f. 

From  these  we  have 

L  sin  (P  -  N)  =  m  sin  (M  -  N), 

(203) 
L  cos  (P  —.N)  =  m  cos  (Jf  —  JV)  -f  ^  r. 

Hence  putting  ^  =  P  —  N,  we  have 

sin  ^  -  mSm(5~^},  (204) 


182  PRACTICAL  ASTRONOMY. 

the  lower  sign  of  the  second  term  in  the  second  member  of  the  last 
equation  corresponding  to  the  time  of  beginning  and  the  upper  to 
the  time  of  ending  of  the  eclipse.* 

4.  The  Position  Angle  of  the  Point  of  Contact.  —  The  angle  re- 

quired is  P  —  N  -f  i/>  for  the  end  and  P  —  N  —  fi  ±  180° 
for  the  beginning  b'f  .the  eclipse. 

5.  We  now  have  all  the  equations/  and  the  Ephemeris  gives  us  the 

Besselian  table  of  elements  from  which  the  circumstances  of 
an  eclipse  can  be  computed  at  any  place.  These  equations 
are  here  arranged  in  the  order  in  which  they  would  be  used, 
and  the  student  is  referred  to  the  type  problem  worked  out 
in  the  Ephemeris  as  a  guide. 

1.  Constants  for  the  given  place, 

p  sin  0'  )  Found  from  table  page  505,  Ephemeris,  know- 
p  cos  0'  f      ing  the  observer's  latitude. 

2.  Coordinates  of  observer,  referred  to  center  of  earth. 

£  =  p  cos  0'  sin  (/*  —  a). 

?/  =  p  sin  0'  cos  d  —  p  cos  0'  sin  d  cos  (/<*  —  #), 

C  =  p  sin  0'  sin  d  -j-  p  cos  0'  cos  d  cos  (yu  —  a). 

3.  Variations  of  observer's  coordinates  in  one  minute  of  mean  time, 

£'  =  [7.63992]  pcos  0'  cos  (//,  -  A). 
7?'  =  [7.63992]  £  sin  d. 

4.  The  values  of  m^  M,  n  and  N,  given  by 

m  sin  M  =  x  —  Z, 
m  cos  M  =  y  —  77, 


n  cos  N=  y'  —  ?/. 


See  page  506,  Ephemeris. 


SOLAR  ECLIP8E.  183 

5.  The  radius  L  of  the  shadow  or  penumbra  on  a  plane  passing 
through  the  observer,  parallel  to  the  fundamental  plane,  and 
at  a  distance  C  from  it. 


6.     The  value  of  the  angle  ^, 

m  sin  (  M  —  N) 

— 


sm  (    = 


7.     The  value  of  the  time  r  in  minutes 

m  cos  (  Jf  —  N)       L 


r  = 


8.     The  position  angle  P,  from 


or 

P  =  N  -  ^  ±  ISO0, 


TABLES. 


<p 

<?—<?' 

log  p 

<P 

<P—<P' 

log  P 

0    1 

i  a 

0    / 

i  a 

0  0 

0  0.00 

0.000  0000 

35  0 

10  48.^5 

9.999  5248 

1  0 

0  24.02 

9.999  9996 

10 

49.63 

5208 

2  0 

0  48.02 

9982 

20 

50.98 

5169 

3  0 

1  11.95 

9961 

30 

52.31 

5129 

4  0 

1  35.80 

9930 

40 

53.62 

5089 

5  0 

1  59.54 

9891 

50 

54.90 

5049 

6  0 

2  23.12 

9.099  9843 

36  0 

10  56.16 

9.999  5009 

7  0 

2  46.54 

9786 

10 

57.41 

4969 

8  0 

3  9.76 

9721 

20 

58.63 

4929 

0  0 

3  82.74 

9648 

30 

59.82 

4888 

10  0 

8  55.47 

9566 

40 

11  1.00 

4848 

11  0 

4  17.92 

9476 

50 

2.15 

4807 

12  0 

4  40.06 

9.999  9377 

37  0 

11  3.28 

9.999  4767 

13  0 

5  1.85 

9271 

10 

4.39 

4726 

14  0 

5  23.28 

9157 

20 

5.47 

4686 

15  0 

5  44.33 

9035 

30 

6.54 

4645 

16  0 

6  4.95 

8905 

40 

7.58 

4604 

17  0 

6  25.14 

8768 

50 

8.59 

4563 

18  0 

6  44.86 

9.999  8624 

88  0 

11  9.59 

9.999  4522 

19  0 

7  4.09 

8472 

10 

10.56 

4481 

20  0 

7  22.80 

8314 

20 

11.51 

4440 

21  0 

7  40.99 

8149 

80 

12.44 

4399 

22  0 

7  58.61 

7977 

40 

13.34 

4358 

23  0 

8  15.66 

7799 

50 

14.22 

4317 

24  0 

8  82.10 

9.999  7614 

89  0 

11  15.08 

9.999  4276 

25  0 

8  47.93 

7424 

10 

15.92 

4234 

20  0 

9  3.12 

7228 

20 

16.73 

4193 

27  0 

9  17.65 

7027 

80 

17.52 

4152 

28  0 

9  31.50 

6820 

40 

18.29 

4110 

29  0 

9  44.66 

6608 

50 

19.04 

4069 

30  0 

9  57.12 

9.999  6392 

40  0 

11  19.76 

9.999  4027 

10 

9  59.12 

6355 

10 

20.46 

3985 

20 

10  1.11 

6319 

20 

21.13 

3944 

80 

3.07 

6282 

80 

21.79 

3902 

40 

5.02 

6245 

40 

22.42 

8860 

CO 

6.94 

6208 

50 

23.02 

8819 

81  0 

10  8.85 

9.999  6171 

41  0 

11  23.61 

9.999  8777 

10 

10.73 

6134 

10 

24.17 

3735 

20 

12.59 

6096 

20 

24.70 

8693 

80 

14.44 

6059 

80 

25.22 

8651 

40 

16.26 

6021 

40 

25.71 

8609 

50 

18.06 

5984 

60 

26.18 

8567 

82  0 

10  19.84 

9.999  5946 

42  0 

11  26.62 

9.999  3525 

10 

21.60 

5908 

10 

27.04 

8483 

20 

23.34 

5870 

20 

27.44 

8441 

80 

25.05 

5832 

eo 

27.82 

8399 

40 

26.75 

5794 

40 

28.17 

8357 

60 

28.43 

5755 

50 

28.50 

8315 

83  0 

10  30.08 

9.999  5717 

43  0 

11  28.80 

9.999  3278 

10 

31.71 

5678 

10 

29.08 

3230 

20 

33.32 

5640 

20 

29.34 

8188 

80 

84.91 

5601 

80 

29.58 

8146 

40 

86.48 

5562 

40 

29.79 

8104 

50 

88.03 

5523 

60 

29.98 

8062 

84  0 

10  89.55 

9.999  5484 

44  0 

11  80.14 

9.999  8019 

10 

41.06 

5445 

10 

80.29 

2977 

20 

42.54 

5406 

20 

80.41 

2985 

80 

44.00 

5367 

80 

80.50 

2892 

40 

45.44 

5327 

40 

80.57 

2850 

50 

46.86 

5288 

50 

80.62 

2808 

187 


I 


I 


i  S 
I  a 


3 

a 

m 

I 


Q. 
P 


188 


DC 

I  I 

UJ  ^ 

**-  .* 

0  I 

E  I 


a.  i 

It 
3  I 


.°  a 

t$  & 

3  h 

•o  * 

<D 

DC 


<p 

<P—<P' 

log/a 

<P 

<P—<p' 

log  p 

0          / 

i     ii 

0          1 

i     a 

45    0 

11  30.«5 

9.999  2766 

55    0 

10  49.74 

9.999  0275 

10 

30.65 

2723 

10 

48.36 

0235 

20 

30.63 

2681 

20 

46.97 

0195 

30 

30.58 

2639 

30 

45.55 

0155 

40 

30.51 

2596 

40 

44.11 

0116 

50 

30.42 

2554 

50 

42.65 

0076 

46    0 

11  80.31 

9.999  2512 

56    0 

10  41.16 

9.999  0037 

10 

80.17 

2470 

10 

39.65 

9.998  9993 

20 

30.01 

2427 

20 

88.13 

9958 

30 

29.82 

2385 

30 

86.58 

9919 

40 

29.61  ' 

2343 

40 

85.01 

9880 

50 

29.38 

2300 

50 

33.41 

9841 

47    0 

11  29.12 

9.999  2258 

57    0 

10  31.80 

9.998  9802 

10 

28.85 

2216 

10 

30.16 

9764 

20 

28.54 

2174 

20 

28.50 

9725 

80 

28.22 

2132 

30 

26.83 

9686 

40 

27.87 

2089 

40 

25.13 

9648 

50 

27.50 

2047 

50 

23.40  . 

9610 

48    0 
10 

11  27.10 
26.69 

9.999  2005 
1963 

58    0 

10 

10  21.66 
19.90 

9.998  9571 
9533 

20 

26.24 

1921 

20 

18.11 

9495 

30 

25.78 

1879 

30 

16.31 

9457 

40 

25.29 

1837 

40 

14.48 

9419 

50 

24.78 

1795 

50 

12.63 

9382 

49    0 

10 
20 
80 
40 
50 

11  24.24 
23.69 
23.11 
22.50 
21.87 
21.22 

9.999  1753 
1711 
1669 
1627 
1586 
1544 

59    0 

10 
20 
80 
40 
50 

10  10.77 
8.88 
6.97 
5.04 
3.08 
1.11 

9.998  9344 
9307 
9269 
9232 
9195 
9158 

60    0 

9  59.12 

9.998  9121 

50    0 

11  20.25 

9.999  1502 

61    0 

9  46.74 

8902 

10 

19.85 

1460 

62    0 

9  33.65 

8688 

20 

19.13 

1419 

63    0 

9  19.85 

6479 

80 

18.39 

1377 

64    0 

9    5.36 

8275 

40 

17.63 

1335 

65    0 

8  50.21 

8077 

50 

16.84 

1294 

66    0 

8  34.40 

9.998  7884 

51    0 

11  16.02 

9.999  1252 

67    0 

8  17.97 

7697 

10 

15.19 

1211 

68    0 

8    0.92 

7517 

20 

14.33 

1170 

69    0 

7  43.29 

7342 

30 

13.45 

1128 

70    0 

7  25.08 

7174 

40 

12.55 

1087 

71    0 

7    6.33 

7013 

50 

11.62 

1046 

72    0 

6  47.06 

9.998  6859 

52    0 
10 

11  10.67 
9.70 

9.999  1005 
0963 

73    0 

74    0 

6  27.28 
6    7.03 

6713 
6573 

20 

8.71 

0922 

75    0 

5  45.33 

6441 

30 

7.69 

0881 

76    0 

5  25.20 

6317 

40 

6.66 

0840 

77    0 

5    3.67 

6201 

50 

5.60 

0800 

78    0 

4  41.77 

t.998  6093 

53    0 

10 
20 
30 
40 

11    4.51 
3.40 
2.27 
1.12 
10  59.94 

9.999  0759 
0718 
0677 
0637 
0596 

7!)    0 

80    0 
81    0 
fc2    0 
83    0 

4  19.53 
3  56.96 
8  34.10 
3  10.98 
2  47.63 

5993 
5901 
-  5818 
£748 
5676 

50 

58.74 

0556 

84    0 

2  24.07 

9.998  5619 

85    0 

2    0.33 

5570 

54    0 

10  57.52 

9.999  0515 

86    0 

1  86.44 

5530 

10 

56.28 

0475 

87    0 

1  12.43 

5498 

20 

55.02 

0435 

88    0 

0  48.34 

5470 

30 

53.73 

OC95 

89    0 

0  24.18 

5463 

40 

52.42 

0355 

50 

51.09 

0315 

90    0 

0    0.00 

9.998  5458 

Apparent 
Altitude. 

Mean 
Refraction. 

Apparent 

Altitude. 

Mean 
Refraction. 

Apparent 
Altitude. 

Mean  Re- 
fractiou. 

Apparent 
Altitude. 

Mean  Re- 
fraction. 

•         / 

/       // 

•       / 

/        n 

•        I 

/         // 

•        / 

/         It 

9    80 

5    85.1 

14    80 

8    41.6 

20      0 

2    38.8 

0      0 

86    29 

85 

5    32.4 

85 

8    40.3 

10 

2    87.4 

1      0 

24    54 

40 

5    29.6 

40 

3    39.0 

20 

2    36.0 

3      0 

18    26 

45 

5    27.0 

45 

3    37.7 

80 

2    34.6 

8      0 

14    25 

50 

5    24.3 

50 

3    36.5 

40 

2    33.3 

4      0 

11    44 

55 

5    21.7 

55 

3    85.8 

60 

2    32.0 

5      0 

9    52.0 

10      0 

5    19.2 

15      0 

3    84.1 

21      0 

2    30.7 

5 

9    44.0 

5 

5    16.7 

6 

3    32.9 

10 

2    29.4 

10 

9    36.2 

10 

5    14.2 

10 

8    81.7 

20 

2    28.1 

15 

9    28.6 

15 

5    11.7 

15 

8    30.5 

80 

2    26.9 

20 

9    21.2 

20 

5      9.3 

20 

8    29.4 

40 

2    25.7 

25 

9     14.0 

25 

5      6.9 

25 

3    28.2 

50 

2    24.5 

30 

9      7.0 

30 

5      4.6 

80 

3    27.1 

22      0 

2    23.3 

35 

9      0.1 

85 

5      2.3 

85 

3    25.9 

10 

2    22.1 

40 

8    53.4 

40 

5      0.0 

40 

3    24.8 

20 

2    20.9 

45 

8    46.8 

45 

4    57.8 

45 

8    23.7 

30 

2    19.8 

50 

8    40.4 

50 

4    55.6 

50 

3    22.6 

40 

2    18.7 

55 

8    34.2 

55 

4    53.4 

65 

3    21.5 

50 

2    17.5 

6      0 

8    28.0 

11      0 

4    51.2 

16      0 

3    20.5 

23      0 

2    16.4 

5 

8    22.1 

5 

4    49.1 

5 

3    19.4 

10 

2    15.4 

10 

8    16.2 

10 

4    47.0 

10 

3    18.4 

20 

2    14.3 

15 

8    10.5 

15 

4    44.9 

15 

8    17.3 

80 

2    13.3 

20 

8      4.8 

20 

4    42.9 

20 

3    16.8 

40 

2     12.2 

25 

7    59.3 

25 

4    40.9 

25 

8    15.2 

50 

2    11.2 

80 

7    53.9 

30 

4    38.9 

30 

3    14.2 

24      0 

2    10.2 

85 

7    48.7 

35 

4    36.9 

35 

8    13.2 

10 

2      9.2 

40 

7    43.5 

40 

4    35.0 

40 

3    12  2 

20 

2      8.2 

45 

7    38.4 

45 

4    33.1 

45 

3    11.2 

30 

2      7.2 

50 

7    33.5 

50 

4    31.2 

50 

8    10.3 

40 

2      6.2 

55 

7    28.6 

55 

4    29.4 

55 

3      9.3 

50 

2      5.3 

7      0 

7    23.8 

12      0 

4    27.5 

17      0 

3      8.3 

25      0 

2      4.4 

5 

7    19.2 

5 

4    25.7 

5 

3      7.3 

10 

2      3.4 

10 

7    14.6 

10 

4    23.9 

10 

3      6.4 

20 

2      2.5 

15 

7    10.1 

15 

4    22.2 

15 

3      5.5 

80 

2      1.6 

20 

7      5.7 

20 

4    20.4 

20 

3      4.6 

40 

2      0.7 

25 

7      1.4 

25 

4    18.7 

25 

3      8.7 

50 

1    59.8 

80 

«    57.1 

30 

4    17.0 

80 

3      2.8 

26      0 

1    58.9 

35 

6    53.0 

85 

4    15.3 

85 

3      1.9 

10 

1     58.1 

40 

6    48.9 

40 

4    13.6 

40 

3      1.0 

20 

1    57.2 

45 

6    44.9 

45 

4    12.0 

45 

8      0.1 

30 

1     56.4 

50 

6    41.0 

50 

4    10.4 

60 

2    59.2 

40 

1    55.5 

55 

6    37.1 

55 

4      8.8 

55 

2    58.3 

60 

1    54.7 

8      0 

6    33.3 

13      0 

4      7.2 

18      0 

2    57.5 

27      0 

53.9 

5 

6    29.6 

5 

4      5.6 

6 

2    56.6 

10 

53.1 

10 

6    25.9 

10 

4      4.1 

10 

2    55.8 

20 

52.3 

15 

6    22.3 

15 

4      2.6 

15 

2    54.9 

80 

6t-.fi 

20 

6    18.8 

20 

4      1.0 

20 

2    54.1 

40 

50.7 

25 

6    15.3 

25 

3    59.6 

25 

2    53.2 

50 

50.0 

30 

6    11.  d 

80 

3    58.1 

30 

2    52.4 

28      0 

1    49.2 

35- 

6      8.5 

35 

3    56.6 

35 

2    51.6 

10 

1    48.4 

40 

e    5.2 

40 

3    55.2 

40 

2    50.8 

20 

1    47.7 

45 

6      2.0 

45 

3    53.7 

45 

2    50.0 

80 

1    46.9 

1         FO 

5    58.8 

50 

3    52.3 

50 

2    49.2 

40 

1     46.2 

I         65 

5    55.7 

55 

3    50.9 

55 

2    48.4 

50 

'1     45.5 

1 

•      0 

5    52.6 

14      0 

3    49.5 

19      0 

2    47.7 

29      0 

1     44.8 

5 

5    49.6 

5 

3    48.1 

10 

2    46.1 

20 

1     43.4 

10 

5    46.6 

10 

3    46.8 

20 

2    44.6 

40 

1     42.0 

15 

5    43.6 

15 

8    45.5 

30 

2    43.1 

30      0 

1     40.6 

20 

6    40.7 

20 

8    44.2 

40 

2    41.6 

20 

1     89.8 

25 

5    87.9 

25 

3    42.9 

50 

2    40.2 

40 

1     38.0 

189 


g 


5-    m 


g  3 

I? 


0  O 

1  3 


190 


Apparent 
Altitude. 

Mean 

Refraction. 

Apparent 
Altitude. 

Mean 
Refraction. 

Apparent 
Altitude, 

Mean  lie- 
fraction. 

Apparent 
Altitude. 

Mean  Re- 
fraction. 

0             1 

81      0 

/          /( 

1    86.7 

•        / 

41      0 

/        // 
1       7.0 

o        / 

51      0 

/         // 

0    47.2 

o        / 
61        0 

/        n 
0    32.3 

20 

1    85.5 

20 

1       6.2 

20 

0    46.6 

62      0 

0    31.0 

40 

1    84.2 

40 

1       5.4 

40 

0    46.1 

63      0 

0    29.7 

82      0 

1    33.0 

42      0 

1      4.7 

58      0 

0    45.5 

64      0 

0    28.4 

20 

1    31.8 

20 

1      8.9 

20 

0    45.0 

65      0 

0    27.2 

40 

1    30.7 

40 

1      8.2 

40 

0    44.4 

66      0 

0    25.9 

88      0 

1    29.5 

48      0 

1      2.4 

53      0 

0    43.9 

67      0 

0    24.7 

20 

1    28.4 

20 

1      1.7 

20 

0    48.4 

68      0 

0    23.6 

40 

1    27.3 

40 

1      1.0 

40 

0    42.8 

69      0 

0    22.4 

84      0 

1     26.2 

44      0 

1      0.3 

54      0 

0    42.8 

70      0 

0    21.2 

20 

1     25.1 

20 

0    89.6 

20 

0    41.8 

71      0 

0    20.1 

40 

1    24.1 

40 

0    58.9 

40 

0    41.3 

72      0 

0    18.9 

85      0 

1    23.1 

45      0 

0    58.2 

55      0 

0    40.8 

78      0 

0    17.8 

20 

1     22.0 

20 

0    57.6 

20 

0    40.3 

74      0 

0    16.7 

40 

1     21.0 

40 

0    66.9 

40 

0    89.8 

75      0 

0    15.6 

86      0 

1     20.1 

46      0 

0    56.2 

56      0 

0    89.3 

76      0 

0    14.5 

20 

1     19.1 

20 

0    55.6 

20 

0    38.8 

77      0 

0     13.5 

40 

1     18.2 

40 

0    55.0 

40 

0    38.3 

78      0 

0    12.4 

87      0 

1    17.2 

47      0 

0    54.8 

57      0 

0    37.8 

79      0 

0    11.3 

20 

1     16.3 

20 

0    53.7 

20 

0    37.8 

80      0 

0    10.8 

40 

1     15.4 

40 

0    53.1 

40 

0    86.9 

81      0 

0      9.2 

38      0 

1     14.5 

48      0 

0    52.5 

58      0 

0    86.4 

82      0 

0      8.2 

20 

1    13.6 

20 

0    51.9 

20 

0    85.9 

88      0 

0      7.2 

40 

1    12.7 

40 

0    51.2 

40 

0    85.5 

84      0 

0      6.1 

89      0 

1     11.9 

49      0 

0    50.6 

59      0 

0    85.0 

85      0 

0      5.1 

20 

1     11.0 

20 

0    50.0 

20 

0    34.5 

86      0 

0      4.1 

40 

1     10.2 

40 

0    49.4 

40 

0    84.1 

87      0 

0      3.1 

40      0 

1      9.4 

50      0 

0    48.9 

60      0 

0    83.6 

88      0 

0      2.0 

20 

1      8.6 

20 

0    48.3 

20 

0    88.2 

89      0 

0      1.0 

40 

1      7.8 

40 

0    47.8 

40 

0    82.7 

90      0 

0      0.0 

191 


Zenith 

dis- 
tance. 

Arg.  app.  zen.-dist. 

Zenith 

dis- 
tance. 

Arg.  app.  zen.-dist 

Log  a. 

A 

A. 

Log  a. 

A 

A 

e  / 

0   1 

0  0 

1.76156  o 

77  0 

1.75229  «. 

1.0026 

1.0252 

10  0 

1.76154  f 

10 

1.75205  5: 

1.0026 

1.0258 

20  0 

1.76149  .,£ 

20 

1.75180  £ 

1.0027 

1.0264 

30  0 

1.76139  Q 

30 

1.75155  X 

1.0027 

1.0272 

35  0 

1.76130  .: 

40 

1.75129  oft 

1.0028 

1.0281 

40  0 

1.76119  Jg 

60 

1.75101  g 

1.0029 

1.0290 

45  0 

1.76104  t 

.0018 

78  0 

1.75072  9Q 

1.0030 

1.0299 

46  0 

1.76100 

.0019 

10 

1.75043  OA 

1.0030 

1.0308 

47  0 

1.76096 

.0019 

20 

1.75013  09 

1.0031 

1.0318 

48  0 

1.76092  * 

.0020 

80 

1.74981  XT 

1.0032 

1.0328 

49  0 

1.76087  R 

.0021 

40 

1.74947  S3 

1.0033 

1.0338 

50  0 

1.76082  g 

1.0023 

50 

1.74912  gg 

1.0034 

1.0347 

51  0 

1.76077  A 

1.0025 

79  0 

1.74876  Q7 

1.0035 

1.0357 

52  0 

1.76071  X 

1.0026 

10 

1.74839  % 

1.0036 

1.0367 

53  0 

1.76065  X 

1.0027 

20 

1.74799  J$ 

1.0037 

1.0377 

54  0 

1.76058  A 

1.0029 

80 

1.74757  To 

1.0038 

1.0387 

55  0 

1.76050 

1.0031 

40 

1.74714  J2 

1.0039 

1.0398 

56  0 

1.76042 

V 

1.0034 

60 

1.74670  JJ 

1.0040 

1.0409 

57  0 

1.76033  in 

1.0037 

80  0 

1.74623  Kn 

1.0041 

1.0420 

58  0 

1.76023  }V 

1.0040 

10 

1.74573  gx 

1.0042 

1.0431 

59  0 

1.76012  *\ 

1.0043 

20 

1.74521  ^ 

1.0043 

1.0442 

60  0 

1.76001  fi 

1.0046 

80 

1.74468  22 

1.0045 

1.0454 

61  0 

1.75988  {£ 

1.0049 

40 

1.74412  £x 

1.0046 

1.0466 

62  0 

1.75973  Jj 

1.0054 

50 

1.74352  gj 

1.0047 

1.0479 

63  0 

1.75957  1ft 

1.0058 

81  0 

1.74288  Af- 

1.0049 

1.0493 

64  0 

1.75939  i£ 

1.0063 

10 

1.74223  *£ 

1.0050 

1.0508 

65  0 

1.75919  ™ 

1.0068 

20 

1.74155  SS 

1.0052 

1.0523 

66  0 

1.75897  ii 

1.0075 

80 

1.74083  '* 

1.0054 

1.0540 

67  0 

1.75871  no 

1.0083 

40 

1.74007  iX 

1.0056 

1.0559 

68  0 

1.75842  g| 

1.0092 

50 

1.73928  gg 

1.0058 

1.0579 

69  0 

1.75809  oo 

1.0101 

82  0 

1.73845  CQ 

1.0060 

1.0600 

70  0 

1.75771  25 

1.0111 

10 

1.73757 

1.0062 

1.0622 

71  0 

1.75726  ™ 

1.0124 

20 

1.73663  X2 

1.0065 

1.0646 

72  0 

1.75675  «i 

1.0139 

80 

1.73564  -xr 

1.0067 

1.0671 

73  0 

1.75615  !£ 

1.0156 

40 

1.73459  fVo 

1.0070 

1.0697 

74  0 

1.75543  •* 

1.0175 

50 

1.73347  Jg 

1.0073 

1.0725 

75  0 

1.75457  1A 

1.0197 

83  0 

1.73229  10. 

10075 

1.0754 

10 

1.75441  }X 

1.0200 

10 

1.78105  }J? 

1.0978 

1,0784 

20 

1.75425  !2 

1.0204 

20 

1.72974  fji 

1.0081 

1.0815 

80 

1.75408  fX 

1.0208 

80 

1.72832  JJf 

1.0084 

1.0846 

40 

1.75391  |A 

1.0212 

40 

1.72681  {fti 

1.0082 

1.0879 

50 

1.75373  Jg 

1.0216 

60 

1.72519  J" 

1.0098 

1.0914 

76  0 

1.75355  1Q 

1.0220 

84  0 

1.72346  1Qft 

1.0096 

1.0951 

10 

1.75336  iX 

1.0225 

10 

1.72160  IQQ 

1.0100 

1.0992 

20 

1.75316  £ 

1.0230 

20 

1.71961  oi« 

1.0105 

1.1036 

30 

1.75295  i\ 

1.0235 

80 

1.71749  Si? 

1.0110 

1.1082 

40 

1.75274  SJ 

1.0241 

40 

1.71522  *Ji 

1.0115 

1.1130 

50 

1.75252  ~* 

1.0246 

50 

1.71279  )*g 

1.0121 

1.1178 

77  0 

1.75229 

1.0026 

1.0252 

85  0 

1.71020 

1.0127 

1.1229 

CD 


P 

cr 


192 


j3 

H 

c 
o 

4-» 

O 

& 

(/) 

"o> 
to 

0) 

00 


UJ 

_l 
m 
< 


Factor  depending    upon                                          Factor  depending  upon  the  external 

the   barometer.                                                                   thermometer. 

Eng.  ln§. 

Log  B.                                   F. 

Log  r- 

F. 

Log  y. 

27.5 

—0.03191                          "       o 

0 

27.6 
27.7 
27.8 
27.9 
28.0 
28.1 
28.2 
28.3 
28.4 
28.5 
28.6 
28.7 
28.8 
28.9 
29.0 
29.1 
29.2 
29.3 
29.4 
29.5 
29.6 
29.7 
29.8 
29.9 
30.0 
80.1 
30.2 
30.3 
30.4 
30.5 
30.6 
30.7 
30.8 
30.9 
31.0 

0.03033                                 2Q 
0.02876 
0.02720                                 ™ 
0.02564                                 :}° 
0.02409                                  fJL 
0.02254                              &  }° 
0-.  02099                                  JJ 
0.01946                                  J* 
0.01793                                  JjJ 
0.01640                     *   ?;       J? 
0.01488                                  H 
0.01336                                   a 
0.01185 
0.01035                                  2 
0.00885                                  L 
0.00735 
0.00586 
0.00438 
0.00290 
—0.00142 
+0.00005 

o.ooisi                     .  y 

0.00297                               !*S 
0.00443 
0.00588 
0.00733 
0.00876 
0.01020                                  2 
0.01163 
0.01306 
0.01448                                -X 
0.01589                                 Jr 
0.01731                                 *i 
0.01871                                 J5 
-fO.02012                                J| 

+  0.06276 
0.06181 
0.06083 
0.05985 
0.05887 
0.05790 
0.05093 
0.05596 
0.05500 
0.05403 
0.05307 
0.05211 
0.05115 
0.05020 
0.04924 
0.04829 
0.04734 
0.04640 
0.04545 
0.04451 
0.04357 
0.04263 
0.04169 
0.04076 
0.03982 
0.03889 
0.03796 
0.03704 
0.03611 
0.03519 
0.03427 
0.03335 
0.03243 
0.03152 
0.03060 

+  35 
86 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 
61 
62 
63 
64 
65 
66 
67  \ 
68 
69 

+0.01185 
0.01098 
0.01011 
0.00924 
0.00837 
0.00750 
0.00004 
0.00578 
0.00492 
0.00406 
0.00320 
0.00234 
0.00149 
+0.00004 
—0.00021 
0.00106 
0.00191 
0.00275 
0.00360 
0.00444 
0.00528 
0.00612 
0.00696 
0.00780 
0.00863 
0.00946 
0.01029 
0.01112 
0.01195 
0.01278 
0.01360 
0.01443 
0.01525 
0.01607 
0.01689 

"                               15 

0.02969 

70 

0.01770 

Factor  depending   nps»                                     jg 

0.02878 

71 

0.01852 

the    attached  ther-                                         j^ 

0.02787 

72 

0.01933 

mometer.                                                  -10 

0.02697 

73 

0.02015 

0.02606 

74 

0.02096 

F. 

Log  T.                                   2Q 

0.02516 

75 

0.02177 

o 

21 

0.02426 

76 

0.02257 

—80 
20 
—10 
0 
10 
+20 
30 
40 
50 
60 
70 
80 
90 
+100 

4-0.00243 
0.00203                                *° 
0.00164                                *3 
0.00125 
0.00086                                  *J 
0.00047                                 ^ 
+000008 
—0.00031                                  £X 
0.00070 
0.00109 
0.00148 
0.00186 
0.00225                               J  gt 
—0.00264 

0.02336 
0.02247 
0.02157 
0.02068 
0.01979 
0.01890 
0.01801 
0.01713 
0.01624 
0.01536 
0.01448 
0.01360 
0.01273 
+0.01185 

77 
78 
79 
80 
81 
82 
83 
84 
85 
86 
87 
88 
89 
+  90 

0.02338 
0.02419 
0.02499 
0.02579 
0.02659 
0.02738 
0.03819 
0.02898 
0.02978 
0.03057 
0.03136 
0.03216 
0.03294 
—0.03373 

Log^=logB-hlogT. 

194 


Elapsed 
Time. 

Log.  A. 

Log.B. 

Elapsed 
Time. 

Log.  A. 

Log.B. 

[Elapsed 
Time. 

Log.  A. 

Log.B. 

/w.   7/». 

h.  m. 

h.  m. 

0   0 

9.4059 

9.4059 

0  40 

9.4065 

9.4048 

1  20 

9.4081 

9.4015 

2 

.4059 

.4059 

42 

.40(55 

.4047 

22 

.4083 

.4013 

4 

.4059 

.4059 

44 

.4066 

.4046 

24 

.4084 

.4010 

6 

.4060 

.4059 

46 

.4067 

.4045 

26 

.4085 

.4008 

8 

.4060 

.4059 

48 

.4067 

.4043 

28 

.4086 

.4006 

10 

.4060 

.4059 

50 

.4068 

'.4042 

30 

.4087 

.4003 

12 

.4060 

.40.18 

52 

.4069 

.4041 

32 

.4089 

.4001 

14 

.4060 

.'4058 

54 

;  .4069 

.4039 

34 

.4090 

.3998 

16 

.4060 

.4058 

56 

.4070 

.4038 

36 

.4091 

.3995 

18 

.4061 

.4057 

0  58 

.4071 

.4036 

38 

.4093 

.3993 

20 

.4061 

.4057 

1   0 

.4072 

.4034 

40 

.4094 

.3990 

22 

.4061 

.4056 

2 

.4073 

.4033 

42 

.4095 

.3987 

24 

.4061 

.4055 

4 

.4074 

.4031 

44 

.4097 

.3984 

26 

.4062 

.4055 

6 

.4074 

.4029 

46 

.4098 

.8981 

28 

.4062 

.4054 

8 

.4075 

.4027 

48 

.4100 

.8978 

30 

.4062 

.4053 

10 

.4076 

.4025 

50 

.4101 

.8975 

82 

.4063 

.4052 

12 

.4077 

.4023 

52 

.4103 

.3972 

34 

.4063 

.4051 

14 

.4078 

.4021 

64 

.4104 

.3969 

36 

.4064 

.4050 

16 

.4079 

.4019 

56 

.4106 

.3965 

0  88 

9.4064 

9.4049 

1  18 

9.4080 

9.4017 

1  58 

9.4107 

0.3962 

Elapsed 
Time. 

Log.  A. 

Log,B. 

Elapsed 
Time. 

Log.  A. 

Log.  B. 

Klapsec 
Time. 

Log.  A. 

Log.  B. 

h.  m. 

h.  m. 

A.  m* 

2   0 

9.4100 

9.3959 

4   0 

9.4260 

9.8635 

6   0 

9.4515 

9.8010 

c 

i 

.4111 

.3955 

2 

.4203 

.8627 

2 

.4521 

.2996 

4 

.4113 

.3952 

4 

.4*66 

.8620 

4 

.4526 

.2982 

e 

.4114 

.8948 

6 

.4270 

.8613 

6 

.4581 

.2968 

8 

.4116 

.3944 

8 

.4273 

.3604 

8 

.4636 

.2954 

10 

.4118 

.8941 

10 

.4277 

.8596 

10 

.4542 

.2940 

12 

.4120 

.8937 

12 

.4280 

.8588 

12 

.4547 

.2925 

14 

.4121 

.8933 

14 

.4284 

.8580 

14 

.4552 

.2911 

16 

.4123 

.3929 

16 

.4288 

.8573 

16 

.4558 

.2896 

18 

.4125 

.3925 

18 

.4291 

.8564 

18 

.4563 

.2881 

80 

.4127 

.8921 

20 

.4295 

.3555 

20 

.4569 

.2866 

22 

.4129 

.8917 

22 

.4299 

.3547 

22 

.4574 

.2850 

24 

.4131 

.3913 

24 

.4302 

.8538 

24 

.4580 

•2835 

26 

.4133 

.8909 

26 

.4306 

.8530 

26 

.4585 

.2819 

28 

.4135 

.3905 

28 

.4310 

.8521 

28 

.4591 

.2804 

80 

.4137 

.3900 

30 

.4314 

.3512 

80 

.4597 

.2788 

82 

.4139 

.3896 

32 

.4317 

.8503 

82 

.4602 

.2772 

84 

.4141 

.8892 

84 

.4821 

.8494 

84 

.4608 

.2756 

86 

.4144 

.3887 

86 

.4325 

.8485 

86 

.4614 

.2789 

88 

.4146 

.3882 

38 

.4329 

.3476 

88 

.4620 

.2723 

40 

.4148 

.3878 

40 

.4333 

.8467 

40 

.4625 

.2706 

42 

.4150 

.3873 

42 

.4337 

.8457 

42 

.4631 

.2689 

44 

.4152 

.3868 

44 

.4341 

.3448 

44 

.4637 

.2673 

46 

.4155 

.3863 

40 

.4345 

.3438 

46 

.4648 

.2655 

48 

.4157 

.3859 

48 

.4349 

.3429 

48 

.4649 

.2688 

50 

.4159 

.3854 

50 

.4353 

.8419 

50 

.4655 

.2620 

52 

.4162 

.3849 

52 

.4357 

.3409 

52 

.4661 

.2602 

54 

.4164 

.3843 

54 

.4381 

.3399 

54 

.4667 

.2584 

56 

.4167 

.3838 

56 

.4366 

.3389 

56 

.4673 

.2566 

2  58 

.4169 

.8833 

4  58 

.4370 

.3379 

6  58 

.4679 

.2548 

8   0 

.4172 

.3828 

5   0 

.4374 

.3369 

7   0 

.4685 

.2530 

2 

.4174 

.3822 

2 

.4378 

.8358 

2 

.4691 

.2511 

4 

.4177 

.3817 

4 

.4383 

.8348 

4 

.4697 

.2492 

6 

.4179 

.3811 

6 

.4387 

.8337 

6 

.4704 

.2473 

8 

.4182 

.8806 

8 

.4391 

.3327 

8 

.4710 

.2454 

10 

.4184 

.3800 

10 

.4396 

.8316 

10 

.4716 

.2434 

12 

.4187 

.3794 

12 

.4400 

.8305 

12 

.4723 

.2415 

14 

.4190 

.3789 

14 

.4405 

.3294 

14 

.4729 

.2395 

16 

.4193 

.8783 

16 

.4409 

.3283 

16 

.4735 

.2375 

18 

.4195 

.8777 

18 

.4414 

.3272 

18 

.4742 

.2355 

20 

.4198 

.3771 

20 

.4418 

.8261 

20 

.4748 

.2334 

22 

.4201 

.3765 

22 

.4123 

.8249 

22 

.4755 

.2318 

24 

.4204 

.3759 

24 

.4427 

.3238 

24 

.4761 

.2292 

26 

.4207 

.8752 

26 

.4432 

.3226 

26 

.4768 

.2271 

28 

.4209 

.3746 

28 

.4437 

.8214 

28 

.4774 

.2250 

80 

.4212 

.3740 

30 

.4441 

.3203 

80 

.4781 

.2228 

82 

.4215 

.3733 

32 

.4446 

.3191 

82 

.4788 

.2206 

34 

.4218 

.8727 

34 

.4451 

.3178 

84 

.4794 

.2184 

36 

.4221 

.3720 

36 

.4456 

.8166 

86 

.4801 

.2162 

88 

.4224 

.3713 

88 

.4460 

.8154 

88 

.4808 

.2140 

40 

.4227 

.3707 

40 

.4465 

.8142 

40 

.4815 

.2117 

42 

.4231 

.3700 

42 

.4470 

.3129 

42 

.4821 

.2094 

44 

.4234 

.3693 

44 

.4475 

.8116 

44 

.4828 

.2070 

46 

.4237 

.3686 

46 

.4480 

.3103 

46 

.4835 

.2047 

48 

.4240 

.8679 

48 

.4485 

.3091 

48 

.4842 

.2023 

50 

.4243 

.3672 

50 

.4490 

.3078 

50 

.4849 

.1999 

52 

.4246 

.3665 

52 

.4494 

.3064 

52 

.4856 

.1974 

54 

.4250 

.3657 

54 

.4500 

.3051 

54 

.4863 

.1950 

56 

.4253 

.8650 

56 

.4505 

.3038 

56 

.4870 

.1925 

8  58 

J.4256 

9.3643 

5  58 

9.4510 

9.3024 

7  58 

9.4877 

9.1900 

195 


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LJ 


Elapsed 
Time. 

Log.  A. 

Log.  B. 

Elapsed 
Time. 

Log.  A. 

Log.  B. 

Ei.vpeedL 

Time.  T-°g-  A- 

Log.  B. 

A  in. 

h.  m. 

A.  m. 

8   0 

9.4884 

9.1874 

14   09.6841 

—9.0971 

16   0 

9.7895 

-9.4884 

2 

.4892 

.1848 

2 

.6856 

.1057 

2 

.7915 

.4937 

4 

.4899 

.1822 

4 

.6872 

.1141 

4 

.7935 

.4990 

6 

.4906 

.1796 

6 

.6887 

.1224 

6 

.7955 

,.5042 

8 

.4913 

.1769 

8 

.6903 

.1306 

8 

.7975 

.5094 

10 

.4921 

.1742 

10 

.6919 

.1387 

10 

.7996 

.5146 

12 

.4928 

.1715 

12 

.6934 

.1468 

12 

.8016 

.5197 

14 

.4935 

.1687 

14 

.6950 

.1547 

14 

.8037 

.5248 

16 

.4943 

.1659 

16 

.6966 

.1625 

16 

.8058 

.5300 

18 

.4950 

.1630 

18 

.6982 

.1703 

18 

.8078 

.5351 

20 

.4958 

.1602 

20 

^6998 

.1779 

20 

.8099 

.5401 

22 

.4965 

.1573 

22 

.7014 

.1855 

22 

.8120 

.5452 

24 

.4973 

.1543 

24 

.7030 

.1930 

24 

.8141 

•  5502 

26 

.4980 

.1513 

26 

.7047 

.2004 

23 

.8162 

.5553 

28 

.4988 

.1483 

28 

.7063 

.2078 

28 

.8184 

.5603 

30 

.4996 

.1453 

30 

.7079 

.2150 

30 

.8205 

.5653 

32 

.5003 

.1422 

32 

.7096 

.2222 

32 

.8227 

.5702 

34 

.5011 

.1390 

34 

.7112 

.2293 

34 

.8248 

.5752 

36 

.5019 

.1359 

36 

.7129 

.2364 

36 

.8270 

.5801 

38 

.5027 

.1327 

38 

.7146 

.2434 

38 

.8292 

.5850 

40 

.5035 

.1294 

40 

.7162 

.2503 

40 

.8314 

.5900 

42 

.5042 

.1261 

42 

.7179 

.2571 

42 

.8336 

.5948 

44 

.5050 

.1228 

44 

.7196 

.2639 

44 

.8358 

.5997 

46 

.5058 

.1194 

46 

.7213 

.2706 

46 

.8380 

.6046 

48 

.5066 

.1159 

48 

.7230 

.2773 

48 

.8402 

.G094 

50 

.5074 

.1125 

50 

.7247 

.2839 

50 

.8425 

.6143 

52 

.5082 

.1089 

52 

.7264 

.2905 

52 

.8447 

.6191 

54 

.5091 

.1054 

54 

.7281 

.2970 

54 

.8470 

.6239 

56 

.5099 

.1017 

56 

.7299 

.3034 

56 

.8493 

.6287 

8  58 

.5107 

.0981 

14  68 

'  .7316 

.3098 

16  58 

.8516 

.6335 

9   0 

.5115 

.0943 

15   0 

.7333 

.3162 

17   0 

.8539 

.6383 

2 

.5123 

.0996 

2 

.7351 

.3225 

2 

.8562 

.6481 

4 

.5132 

.0867 

4 

.7369 

.3287 

4 

.8585 

.6478 

6 

.5140 

.0828 

6 

.7386 

.3350 

6 

.8608 

.6526 

8 

.5148 

.0789 

8 

.7404 

.3411 

8 

.8632 

.6573 

10 

.5157 

.0749 

10 

.7422 

.3472 

10 

.8655 

.6621 

12 

.5165 

.0708 

12 

.7440 

.3533 

12 

.8679 

.6668 

14 

.5174 

.0667 

14 

.7458 

.3593 

14 

.8703 

.6715 

16 

.5182 

.0625 

16 

.7476 

.3653 

16 

.8727 

.6762 

18 

.5191 

.0583 

18 

.7494 

.3713 

18 

.8751 

.6809 

20 

.5199 

.0540 

20 

.7512 

.3772 

20 

.8775 

.6856 

22 

.5208 

.0496 

22 

.7531 

.3831 

22 

.8799 

.  6903 

24 

.5217 

.0452 

24 

.7549 

.3889 

24 

-8824 

.6949 

26 

.5225 

.0406 

26 

.7568 

.3947 

26 

.8848 

.6996 

28 

.5234 

.0360 

28 

.7586 

.4005 

28 

.8873 

.7043 

30 

.5243 

.0314 

30 

.7605 

.4062 

30 

.8898 

.7089 

32 

.5252 

.0266 

32 

.7924 

.4119 

32 

.8923 

.7136 

84 

.5261 

.0218 

34 

.7642 

.4175 

34 

.8948 

.7182 

36 

.5269 

.0169 

36 

.7661 

4232 

36 

.8973 

.7228 

38 

.5278 

.0119 

38 

.7680 

]4288 

38 

.8999 

.7275 

40 

.5287 

.0069 

40 

.7699 

.4343 

40 

.9024 

.7321 

42 

.5296 

.0017 

42 

.7718 

.4399 

42 

.9050 

.7367 

44 

.5305 

8.9965 

44 

.7738 

.4454 

44 

.9075 

.7413 

46 

.5315 

.9911 

46 

.7757 

.4509 

46 

.9101 

.7459 

48 

.5324 

.9857 

48 

.7776 

.4563 

48 

.9127 

.7505 

50 

.5333 

.9802 

50 

.7796 

.4617 

50 

.9154 

.7553 

52 

.5342 

.9745 

52 

.7815 

.4671 

52 

.9180 

.751'S 

54 

.5351 

.9688 

54 

.7835 

.4725 

54 

.9206 

.7644 

56 

.5361 

.9630 

56 

.7855 

.4778 

56 

.9233 

.7690  ' 

9  5b 

9.5370 

8.9570 

15  58 

9.7875 

-9.4831 

17  58 

9  .  9260 

—9.7736 

1  Elapsed 
Time. 

Log.  A. 

Log.  B. 

Elapsed 
Time. 

Log.  A. 

Leg.  B. 

Elapsed 
Time. 

Log.  A. 

Log.  B. 

h.  m. 

h.  m. 

h.  m. 

18   0 

9.9287 

—9.7782 

20   0 

0.1249 

—0.0625 

22   0 

0.4523 

—0.4373 

21  .9314 

.7827 

2 

.1290 

.0676 

2 

.4601 

.4455 

4!  .9341 

.7873 

4 

.1330 

.0727 

4 

.4680 

.4540 

6 

.9368 

.7919 

6 

.1371 

.0779 

6 

.4761 

.4625 

8 

.9396 

.7235 

8 

.1412 

.08CO 

8 

.4842 

.4711 

10 

.9424 

.8011 

10 

.1454 

.0882 

10 

.4926 

.4799 

12 

.9451 

.8057 

12 

.1496 

.0935 

12 

.5010 

.48*0 

14 

.9479 

.8103 

14 

.1538 

.0987 

14 

.5C97 

.4980 

16 

.9508 

.8149 

16 

.1581 

.1040 

16 

.5184 

.5072 

18 

.9536 

.8195 

18 

.1623 

.1093 

18 

.5274 

.5165 

20 

.9564 

.8241 

20 

.1667 

.1146 

20 

.5365 

.5261 

22|  .9393 

.8287 

22 

.1711 

.1200 

oo 

.5458 

.5358 

24 

.9622 

.8333 

24 

.1755 

.1253 

24 

.5553 

•5457 

26 

.9651 

.8379 

26 

.1799 

.1308 

26 

.5649 

.5557 

28 

.9680 

.8425 

28 

.1844 

.1862 

28 

.5748 

.5660 

80 

.9709 

.8471 

30 

.1889 

.1417 

30 

.5848 

.5764 

82 

.9739 

.8517 

32 

.1935 

.1472 

82 

.5951 

.5871 

»  34 

.9769 

.8563 

34 

.1981 

.1527 

84 

.6056 

.5979 

86 

.9798 

.8609 

36 

.2028 

.1582 

36 

.6164 

.601)0 

88 

.9829 

.8655 

38 

.2075 

.1638 

88 

.6273 

.6204 

40 

.9859 

.8701 

40 

.2122 

.1695 

40 

.6380 

.6319 

42 

.9889 

.8748 

42 

.2170 

.1751 

42 

.6501 

.64C8 

44 

.9920 

.8794 

44 

.2218 

.1808 

44 

.6619 

.6559 

46 

.9951 

.8840 

46 

.2267 

.1863 

48 

.6740 

.6684 

48 

9.9982 

.8887 

48 

.2316 

.1924 

48 

.6865 

.6811 

60 

0.0013 

.8933 

50 

.2366 

.1982 

50 

.6993 

.6942 

52 

.0044 

.8980 

52 

.2416 

.2040 

52 

.7124 

.7076 

64 

.0076 

.9026 

54 

.2467 

.2099 

54 

.7259 

.7214 

56 

.0108 

.9073 

56 

.2518 

.2159 

56 

.7398 

.7355 

18  58 

.0140 

.9120 

20  53 

.2570 

.2219 

22  58 

.7541 

.7501 

19   0 

.0172 

.9167 

21   0 

.2623 

.2279 

23   0 

.7689 

.7652 

2 

.0204 

.9213 

2 

.2676 

.2339 

2 

.7842 

.7807 

4 

.0237 

.9260 

4 

.2729 

.2401 

4 

.8000 

.7967 

6 

.0270 

.9307 

6 

.2783 

.2462 

6 

.8163 

.8133 

8 

.0303 

.9355 

8 

.2838 

:2524 

8 

.8333 

.8305 

10 

.0336 

.9402 

10 

.2893 

.2587 

10 

.8508 

.8483 

12 

.0370 

.9449 

12 

.2949 

.2650 

12 

.8691 

.8667 

14 

.0403 

.9497 

14 

.3005 

.2714 

14 

.8882 

.886Q 

16 

.0437 

.9544 

16 

.3063 

.2778 

16 

.9080 

.9060 

18 

.0472 

.9592 

18 

.3120 

.2843 

18 

.9288 

.9270 

20 

.0506 

.9640 

20 

.3179 

.2909 

20 

.9506 

.9489 

22 

.0541 

.9687 

22 

.3238 

.2975 

22 

.9734 

.9719 

24 

.0576 

.9735 

24 

.3298 

.3041 

24 

0.9975 

—0.9961 

26 

.0611 

.9784 

26 

.3359 

.3109 

26 

1.0228 

-1.0216 

28 

.0646 

.9832 

28 

.3420 

.3177 

28 

.0497 

4)487 

80 

.0682 

.9880 

30 

.3482 

.3245 

30 

.0783 

.0774 

82 

.0718 

.9929 

32 

.3545 

.3315 

32 

.1089 

.1081 

84 

.0754 

—9.9977 

34 

.3609 

.3385 

34 

.1416 

.1409 

36 

.0790 

—0.0026 

36 

.3674 

.8456 

36 

.1770 

.1764 

88 

.0827 

.0075 

38 

.3739 

.3527 

38 

.2154 

.2149 

40 

.0864 

.0124 

40 

.3805 

.3599 

40 

,2573 

.2569 

42 

.0901 

.0173 

42 

.3873 

.3673 

42 

.3037 

.3033 

44 

.0939 

.0223 

44 

.3941 

.3747 

44 

.3554 

.3553 

46 

.0976 

.0272 

46 

.4010 

.8822 

46 

.4140 

.4138 

48 

.1015 

,0322 

48 

.4080 

.8897 

48 

.4815 

.4814 

50 

.1053 

.0372 

50 

.4151 

.3974 

50 

.5613 

.5612 

52 

.1092 

.0422 

52 

.4223 

.4052 

52 

.6588 

.6587 

54 

.1131 

.0473 

54 

.4297 

.4130 

64 

.7844 

.7843 

56 

.1170 

.0523 

56 

.4371 

.4210 

56 

1.9610 

—1.9610 

19  58 

0.1209 

—0.0574 

21  58 

0.4446 

-0.4291 

23  58 

2.2627 

—2.2627 

197 


M  CD 

o  r 

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I  5s 


» 


2.   3 


b  > 


CO 


p 

0™ 

lm 

2- 

3m 

£» 

5m 

Om 

7m 

g» 

8 

ii 

// 

n 

// 

II 

n 

// 

/  / 

// 

0 

0.00 

1.96 

7.85 

17.67 

31.42 

49.09 

70.68 

96.20 

125.65 

1 

0.00 

2.03 

7.98 

17.87 

31.68 

49.41 

71.07 

96.66 

126.17 

2 

0.00 

2.10 

8.12 

18.07 

31.94 

49.74 

71.47 

97.12 

126.70 

3 

0.00 

2.16 

8.25 

18.27 

32.20 

50.07 

71.86 

97.58 

127.22 

4 

0.01 

2.23 

8.39 

18.47 

32.47 

50.40 

72.26 

98.04 

127.75 

5 

0.01 

2.31 

8.52 

18.  G~ 

32.74 

50.73 

72.66 

98.50 

128.28 

0 

0.02 

2.38 

8.66 

18.87 

83.01 

51.07 

73.06 

98.97 

128.81 

7 

0.02 

2.45 

8.80 

19.07 

33.27 

51.40 

73.46 

99.43 

129.34 

8 

0.03 

2.52 

8.94 

19.28 

33.54 

51.74 

73.86 

99.90 

129.87 

1) 

0.04 

2.60 

9.08 

19.48 

33.81 

52.07 

74.26 

100.37 

130.40 

10 

0.05 

2.67 

9.22 

'  .19.69 

*  34.09 

52.41 

74.66 

100.84 

130.94 

11 

0.06 

2.75 

9.36 

19.90 

;    34.36 

52.75 

75.06 

101.31 

131.47 

12 

0.08 

2.83 

9.50 

20.11 

34.64 

53.09 

75.47 

101.78 

182.01 

13 

0.09 

2.91 

9.64 

20.32 

34.91 

53.43 

75.88 

102.25 

132.55 

14 

0.11 

2.99 

9.79 

20.53 

35.19 

53.77 

76.29 

102.72 

133.09 

15 

0.12 

3.07 

9.94 

20.74 

35.46 

54.11 

76.69 

103.20 

133.63 

10 

0.14 

8.15 

10.09 

20.95 

35.74 

54.46 

77.10 

103.67 

134.17 

17 

0.16 

3.23 

10.24 

21.16 

36.02 

54.80 

77.51 

104.15 

134.71 

18 

0.18 

3.32 

10.39 

21.38 

36.30 

55.15 

77,93 

104.63 

135.25 

11) 

0.20 

3.40 

10.54 

21.60 

36.58 

55.50 

78.84 

105.10 

135.80 

20 

0.22 

3.49 

10.69 

21.82 

36.87 

55.84 

78.75 

105.58 

136.34 

21 

0.24 

3.58 

10.84 

22.03 

37.15 

56.19 

79.16 

106.06 

186.88 

22 

0.26 

3.67 

11.00 

22.25 

37.44 

56.55 

79.58 

106.55 

137.43 

23 

0.28 

3.76 

11.15 

22.47 

37.72 

56.90 

80.00 

107.03 

137.98 

24 

0.31 

8.85 

11.81 

22.70 

38.01 

57.25 

80.42 

107.51 

138.53 

25 

0.34 

3.94 

11.47 

22.92 

38.30 

57.60 

80.84 

107.99 

139.08 

20 

0.87 

4.03 

11.63 

23.14 

38.59 

57.96 

81.26 

108.48 

139.63 

27 

0.40 

4.12 

11.79 

23.37 

38.88 

58.32 

81.68 

108.97 

140.18 

28 

0.43 

4.22 

11.95 

28.60 

39.17 

58.68 

82.10 

109.46 

140.74 

20 

0.46 

4.32 

12.11 

23.82 

39.46 

59.03 

82.52 

109.95 

141.29 

30 

0.49 

4.42 

12.27 

24.05 

89.76 

59.40 

82.95 

110.44 

141.85 

31 

0,52 

4.52 

12.43 

24.28 

40.05 

59.75 

83.38 

110.93 

142.40 

32 

0.56 

4.62 

12.60 

24.51 

40.35 

60.11 

83.81 

111.43 

142.96 

33 

0.59 

4.72 

12.76 

24.74 

40.65 

60.47 

84.23 

111.92 

143.52 

34 

0.63 

4.82 

12.93 

24.98 

40.95 

60.84 

84.66 

112.41 

144.98 

35 

0.67 

4.92 

13.10 

25.21 

41.25 

61.20 

85.09 

112.90 

144.64 

30 

0.71 

5.03 

13.27 

25.45 

41.55 

61.57 

85.52 

113.40 

145.20 

87 

0.75 

6.13 

13.44 

26.68 

41.85 

61.94 

85.95 

11390 

145.76 

38 

0.79 

5.24 

13.62 

25.92 

42.15 

62.81 

86.39 

114.40 

146.33 

3U 

0.83 

5.84 

13.79 

26.16 

42.45 

62.68 

86.82 

114.90 

146.89 

40 

0.87 

5.45 

13.96 

26.40 

42.76 

63.05 

87.26 

115.40 

147.46 

41 

0.91 

6.56 

14.13 

26.64 

43.06 

63.42 

87.70 

115.90 

148.03 

42 

0.96 

6.67 

14.31 

26.88 

43.37 

63.79 

88.14 

116.40 

148.60 

4,'J 

1.01 

5.78 

14.49 

27.12 

43.68 

64.16 

88.57 

116.90 

149.17 

44 

1.06 

5.90 

14.67 

27.87 

43.99 

64.54 

89.01 

117.41 

149.74 

45 

.10 

6.01 

14.85 

27.61 

44.30 

64.91 

89.45 

117.92 

150.31 

40 

.15 

6.13 

15.03 

27.86 

44.61 

65.29 

89.89 

118.43 

150.88 

47 

.20 

6.24 

15.21 

28.10 

44.92 

65.67 

90.33 

118.94 

151.45 

48 

.26 

6.36 

15.39 

28.35 

45.24 

66.05 

90.78 

119.45 

152.03 

49 

.31 

6.48 

15.57 

23.60 

45.55 

66.43 

91.23 

119.96 

152.61 

50 

.36 

6.60 

15.76 

28.85 

45.87 

66.81 

91.68 

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0.49 
0.52 

25    0 

10 

20 

3.64 
3.74 

3.84 

0 
+  1 

0.000  0000 
0101 
0201 

33 

1382.5 

1488.5 

1598.3 

1712.1 

30 

0.54 

30 

3.94 

V^V/X 

nono 

31 

1384.2 

1490.3 

1600.2 

1714.0 

40 

0.50 

40 

4.05 

4 

VOvnO 

0402 

35 

1385.9 

1492.1 

1602.1 

1715.9 

50 

0.59 

50 

4.15 

5 

0503 

3(> 

1387.7 

1493.9 

1604.0 

1717.9 

16    0 

0.61 

26    0 

4.20 

6 

0603 

37 

1389.4 

1495.7 

1605.9 

1719.8 

10 

0.64 

10 

4.37 

7 

0704 

38 

1391.2 

1497.5 

1607.7 

1721.7 

20 

0.67 

20 

4.48 

8 

0804 

39 

1392.9 

1499.3 

1609.6 

1723.6 

30 

0.69 

30 

4.00 

9 

0905 

40 

0.72 

404.72 

40 

1394.7 

1501.1 

1611.5 

1725.6 

50 

0.75 

50 

4.83 

10 

1005 

41 

1396.4 

1502.9 

1613.3 

1727.5 

11 

1106 

42 

1398.2 

1504.7 

1615.2 

1729.5 

17    0 

0.78 

37   04.90 

12 

1206 

43 

1399.9 

150(5.5 

1617.1 

1731.5 

10 

0.81 

105.08 

13 

1307 

44 

1401.7 

1508.4 

1019.0 

1733.4 

20 

0.84 

20  5  .  20 

14 

1407 

30 

0.88 

305.33 

45 

1403.4 

1510.2 

1620.8 

1735.3 

40 

0.91 

405.46 

15 

1508 

4(> 

1405.2 

1512.0 

1622.7 

1737.2 

50 

0.95 

505.60 

16 

1608 

47 

48 
49 

1406.9 

1408.7 
1410.4 

1513.8 
1515.6 
1517.4 

1624.6 
1626.5 
1628.3 

1739.2 
1741.2 
1743.1 

18    0 
10 
20 

0.98 
1.02 
1.06 

28   0 
10 
20 

5.73 

5.87 
6.01 

17 
18 
19 

-  1709 
1809 
1910 

50 
51 
52 

1412.2 
1413.9 
1415.7 

1519.2 
1521.0 
1522.9 

1630.2 
1(332.1 
1634-0 

1745.1 
1747.0 
1749.0 

30 
40 
50 

1.09 
1.13 
1.18 

30J6.15 
40.0.30 

506.44 

20 
21 
22 

OQ 

2010 
2111 
2211 

CO-JO 

53 
54 

1417.4 
1419.2 

1524.7 
1526.5 

1635.9 
1637.7 

1750.9 

1752.8 

19    0 

10 

1.22 
1.26 

29    0 

10 

6.59 
6.75 

ISO 

24 
OK 

£dl,& 

2412 

OK1Q 

55 

5<> 
57 

58 

1420.9 
1422.7 
1424.4 
1426.2 

1528.3 
1530.2 
1532.0 
1533.8 

1639.6 
1641.5 
1643.3 
1645.2 

1754.8 
1756.8 

1758.7 
1700.7 

20 
30 
40 
50 

1.80 
1.3.) 
1.40 
1.44 

20 
30 
40 
60 

6.90 
7.06 
7.22 
7.88 

0U 

26 
27 
28 
29 

jCilxO 

2613 
2714 
2814 
2915 

59 

1427.9 

1535.6 

1647.1 

1702.6 

20    0 

1.49 

30   0 

7.5-r>:     _>_;.{() 

0.000  3015 

201 


33 

CD 

a 
c 
o 

6' 

3 


CD 


202 


0) 
-0 


C 
0) 
s- 
flj 
Q. 

a 


-:      o 


00 


E 

0) 

CO 

0) 

•I 

O 
O 


c 
a) 

bO 


Appar- 
ent 
Alti- 
tude. 

Horizontal  Semi-Diameter. 

i     n 
14  30 

/    // 
15  0 

/     // 
15  30 

/    // 
16  0 

/     // 
16    30 

;     // 
17    0 

// 

n 

n 

// 

ii 

n 

0 

0.10 

0.12 

0.13 

0.14 

0.15 

0.17 

2 

0.58. 

0.62 

0.66 

0.71 

0.76 

0.81 

4 

1.05 

1  1.12 

7-1.20 

1.28 

1.37 

1.46 

6 

1.51 

1.62 

'1-74 

1.86 

1.98 

2.10 

8 

1.98 

2.12 

2.27 

2.42 

2.58 

2.75 

10 

2.44 

2.62 

2.80 

2.99 

3.18 

3.39 

12 

2.90 

3.11 

3.33 

3.56 

3.78 

4.02 

14 

3.36 

3.61 

3.86 

4.11 

4.37 

4.66 

16 

3.82 

4.10 

4.38 

4.67 

4.97 

5.28 

18 

4.28 

4.58 

4.89 

5.22 

5.56 

5.90 

20 

4.72 

5.06 

5.41 

5.76 

6.14 

6.52 

22 

5.16 

5.53 

5.91 

6.30 

6.71 

7.13 

24 

5.60 

5.99 

6.41 

6.83 

7.27 

7.72 

26 

6.03 

6.45 

6.90 

7.35 

7.83 

8.31 

28 

6.45 

6.91 

7.38 

7.87 

8.37 

8.89 

80 

6.86 

7.35 

7.85 

8.87 

8.91 

9.46 

32 

7.27 

7.78 

8.32 

8.87 

9.44 

10.02 

34 

7.67 

8.21 

8.77 

9.35 

9.95 

10.57 

36 

8.06 

8.62 

9.22 

9.83 

10.46 

11.11 

38 

8.43 

9.03 

9.65 

10.29 

10.95 

11.63 

40 

8.80 

9.42 

10.07 

10.74 

11.43 

12.14 

42 

9.16 

9.80 

10.48 

11.17 

11.89 

12.63 

44 

9.51 

10.17 

10.88 

11.60 

12.34 

13.11 

46 

9.84 

10.54 

11.26 

12.01 

12.78 

13.57 

48 

10.16 

10.88 

11.63 

12.40 

13.20 

14.02 

50 

10.48 

11.22 

11.99 

12.78 

13.60 

14.45 

52 

10.78 

11.54 

12.33 

13.15 

13.99 

14.86 

54 

11.07 

11.84 

12.65 

13.50 

14.36 

15.25 

56 

11.34 

12.14 

12.97 

13.83 

14.72 

15.63 

58 

11.60 

12.42 

13.27 

14.15 

15.05 

15.99 

60 

11  84 

12.68 

13.55 

14.44 

15.87 

16.82 

62         12.07 

12.93 

13.81 

14.73 

15.67 

16.64 

64 

12.29 

13.16 

14.06 

14.99 

15.95 

16.94 

66 

12.49 

13.37 

14.29 

15.24 

16.21 

17.22 

68 

12.68 

13.58 

14.50 

15.46 

16.45 

17.47 

70 

12.85 

13.76 

14.70 

15.67 

16.67 

17.71 

72 

13.00 

13.92 

14.88 

15.86 

16.88 

17.92 

74 

13.14 

14.07 

15.04 

16.03 

17.06 

18.12 

76 

13.27 

14.21 

15.18 

16.18 

17.22 

18.29 

78 

13  38 

14.32 

15.30 

16.31 

17.36 

18.43 

80 

13.47 

14.42 

15.40 

16.42 

17.47 

18.58 

82 

13.54 

14.50 

15.49 

16.51 

17.57 

18.66 

84 

13.60 

14.56 

15.56 

16.59 

17.65 

18.74 

86 

13.64 

14.61 

15.60 

16.64 

17.70 

18.80 

88 

18.67 

14.63 

15.63 

16.67 

17.73 

18.83 

90 

13.67 

14.63 

15.64 

16.68 

17.74 

18.85 

FORMS. 


FORMS. 

FOKM  No.  1. 


205 


ERROR  OF  SIDEREAL  TIME-PIECE  BY  MERIDIAN  TRANSIT 

OF  bTAR. 

Station,  WEST  POINT,  N.  Y.    Latitude,  41°  23' 22". 11.     Chronometer  No ,  by 


Date: 

Observer. 

Recorder. 

Transit. 

Illumination. 

Name  of  Star. 

(  Direct. 

E.      W. 

E.      W. 

E.      W. 

E.      W. 

Level.    -^ 
(  Reversed. 

E.      W. 

E.      W. 

E.      W. 

E.      W. 

h     m     s 

h    m    s 

h    m    s 

h    m    s 

.  fl. 

|   n. 

|  in. 

1  -1  IV- 

£     V. 

o 

H  [VII. 

Sum. 

Mean. 

Red.  to  Mid.  Wire. 

Cliron.   Time  of  Transit  over  Mid.   ) 
Wire  =  T.                                          ) 

Level  Error  =  6. 

Level  Correction  =  £6. 

t 

Collimation  Error  =  c. 

Collimation  Correction  =  Cc. 

Azimuth  Error  =  a. 

Azimuth  Correction  =  Aa. 

Chron.  Time  of  Transit. 

App.  R.  A.  of  Star  =  a. 

Error  of  Chrou.  =  E. 

206 


FORMS. 

FORM  No.  2. 


ERROR  OF  MEAN-SOLAR  TIME-PIECE  BY  MERIDIAN  TRANSIT 

OF  SUN. 


Date. 

Latitude  41°  23' 82".  11. 

Observer. 

Transit  No By 


Station,  WEST  POINT,  N.  Y. 

Longitude  4.93A. 

Recorder. 

Mean-Solar  Chron.  No By 


Chronometer  Time  of  Transit  of  West  Limb.      Wire  I 

44  II 

44  III 

"  IV 

44  V 

44  VI 

44  VII 

Chronometer  Time  of  Transit  of  East  Limb.          4t  I 

44  II 

44  III 

44  IV 

44  V 

44  VI 

44  VII 


SUM.          _. 

Chron.  Time  of  Transit  of  Center  over  Mean  of 

Wires  —  Mean.  

Reduction  to  Middle  Wire.  

Level      Error Level  Correction.  

Col.  44    Col.  44 

Azimuth     4t     Azimuth      "  _. 

Chronom.  Time  of  App.  Noon.  

Apparent      '4      44      44         "  12h 0.0m 0.0s... 

Eq.  of  Time. 

Mean  Time  of  Apparent  Noon.  

Error  of  Chronometer  on  Mean  Solar  Time  at 

App.  Noon.  


FORMS.  207 

FOKM  Nc.  3. 

ERROR  OP  SIDEREAL  TIME-PIECE  BY    SINGLE    ALTITUDE    OF 
STAR.      NAME. 

Date.  Station,  WEST  POINT,  N.  Y. 

Observer.  Recorder. 

Sextant  No By Sidereal  Chronom.  No Bv 

•          '  "  T         rjT~ a 

Observed  Double  Altitude.  Chronom.  Time. 


Mean  Sum 

IndexError.  „ t  Mean  =  1 0 


Eccentricity.  Barometer 

Corrected  Double  Altitude Att.  Thermom 

"        Altitude=a8.  Ext.         " 

*Ref  raction  =  r. „ Refraction 


True  Altitude  =a  ., 

Latitude  =£.                                   ..41°..23'..22".ll..         a.  c.  log  cos  <£ 
N.  Polar  Dist.  =cZ.  "     "    sin  d 


cos  m 

sin  (m  -  a) 


"    siniP 

iP 

P 

P  in  Time 

Apparent  R.  A.  of  Star 
Sidereal  Time  =  R.  A.  -j-  P 
Mean  of  Chron.  Times  =•£„ 
Error  of  Chronometer 


*  The  correction  to  be  added  to  this  value  of  r,  if  desired   (see  Note  3,  Text),  is 

2  sin*  a   2     S"n  sin°l^~ '  A  denotin£  the  different  corrected  altitudes,  a0  their  mean,  and 

n  the  number  of  observations.    The  values  of  —       .  n°f~ — -  are  taken  from  Tables  (first 

converting  a0  —  A  into  its  equivalent  in  time),  as  explained  under  "  Latitude  by  Circura- 
Meridian  Altitudes." 

t  The  correction  to  be  added  algebraically  to  this  value  of  t0  if  desired  (see  Note  3, 

Text)  is,  after  computing  Pin  arc,  ^fcot  P  -  gULJ! ;>os  ^  sin  rf"|  2  ifinlj.(^- ^  ^being 

15L  cos  a  cot  a      J  n  sin  1" 

the  different  chronometer  times.    The  last  factor  is  taken  from  Tables  as  before. 


208 


FORMS. 

FORM  No.  4. 


ERROR  OF   MEAN-SOLAR    TIME-PIECE    BY    SINGLE    ALTITUDE 
OF  SUN'S..  ..LIMB. 


Date. 
Observer. 
Sextant  No . 


By. 


Station,  WEST  POINT,  N.  Y, 

Recorder. 

sM.  S.  Chronorn.  No By .  .*„. 


Observed  Double  Altitude. 


Chronom.  Time. 


h  m  s 


Mean 
Index  Error. 


Sum 
tMean  =  tn 


Eccentricity. Barometer 

Att.  Thermom. 
Corrected  Double  Altitude o........  Ext 

"         Altitude  =  «0 Refraction 


*Refraction  =  r. 
Semi-diameter. 
Apparent  Altitude  =  a'. 

Parallax  in  Altitude. 
True  Altitude  =  a. 
Latitude  =  <£. 

N.  Polar  Dist.  =  d. 
a  4-  <t>  +  d 


Longitude  =  4.931  hours. 
Assumed  Error  of  Chronom.  = 
Resulting  Greenwich  Time  of  Obs.: 


„ .     Log.  Eq.  Hor.  Parallax 

1     P. 

"     cos  a'. 

..41°..23'..22".ll..  Parallax  in  Altitude. 


Dec.  at  Greenwich.  Mean  Moon. 
Hourly  Change  X  Greenwich  Time 
Sun's  Declination. 


a.  c.  log  cos  (/> 
"      "    sin  d 
log.  cos  m 
"    sin  (m  —  a) 
"    sin2  i  P 
P 

Pin  Time 
Apparent  Time 
Equation  of  Time. 
Mean  Time. 

Mean  of  Chron.  Times  =  f0 
Error  of  Chronometer. 


*  See  foot-note  to  Form  3. 
t    "       "        "    "       "    " 

NOTE.— For  correction  of  Semi-diameter  due  to  difference  of  refraction  between  limb 
and  center,  see  "  Longitude  by  Lunar  Distances." 


FORMS. 

FOEM  No.  5. 


209 


ERROR  OF  SIDEREAL  TIME-PIECE  BY  EQUAL  ALTITUDES  OF 

A  STAR. 


Station,  WEST  POINT,  N.  Y. 

Observer 

Sextant  No By 

Name  of  Star.  , 


Latitude,  41°  23'  22".ll  =  <f>. 

Recorder 

Sid.  Chronom.  No    By 

App.  Declination  =  8 


Observations  East. 
Observed  Double  Altitudes. 

o  I  tff 


Date 


Chronometer  Times. 
h        m        s 


I. 

II. 
III. 


Barom. 

Att.  Thermom. 

Ext. 

1st  Refraction 


Mean  =  2a. 


Sum 


(Correct  this  for  index  error,  if  correction     1st  Mean, 
for  refraction  be  taken  into  account.) 


Observations  West. 
Observed  Double  Altitudes. 

o  I  If 


Date 


Chronometer  Times. 
h        m        s 


I. 
H. 
III. 


Mean  =  2a. 


Barom. 

Att.  Thermom. 

Ext. 

2d  Refraction 

.1st 
Difference 


(Same  as  above). 


Elapsed  Time. 

l&  Elapsed  Time  in  arc  =  t. 


Middle  Chronometer  Time. 

Correction  for  Refraction. 

Chronom.  Time  of  Transit. 

App.  R.  A.  of  Star. 

Error  of  Chronom.  at  Time  of  Transit. 


Sum   

2d  Mean Log  Difference 

1st     "      Log  cos  a 

a.  c.  log  30 

a.  c.  log  cos  <£ 

...o a.  c.  log  cosS 

a.  c.  log  sin  t 

Log  Correction 

Correc  tion 


210 


FORMS. 


FORM  No.  6. 


ERROR  OF  MEAN-SOLAR  TIME-PIECE   BY   EQUAL   ALTITUDES 
OF  SUN'S LIMB. 

Station,  WEST  POINT,  N.  Y.          <f>  =  Latitude,  41°  23'  22"  11.  Longitude  4.931A,  west. 

Observer y  Recorder . 

'    Sextant  No By M.J&  Chronom.  No By 

Sun's  App.  Dec.  at  local  App.  Noon  (or  midnight)  =  &  = 

Hourly  change  in  &  at  same  time,  =  k  — 

Observations  East.  Date 

Observed  Double  Altitudes.  Chronometer  Times. 

»             •             "  h        m        s 

L  Barom.  

IL  Att.  Thermom 

III.  Ext.          "  

1st  Refraction     

Mean  =  2u Sum  . 

(Correct  this  for  index  error,  if  correction    1st  Mean , 

for  refraction  be  taken  into  account). 

Observations  West.  Date 

Observed  Double  Altitudes.  Chronometer  Times. 

0  '  "  fi        m        s 

I.  Barom.  

II.  Att.  Thermom 

m.  Ext.       "        

2d  Refraction       .... 

1st  "  . 

Mean  =  2a Sum    Difference  

(Same  as  above).  2d  Meai Log  Difference    

1st     "      Log  cos  a  

a.  c.  lag  30  

Elapsed  Time a.  c.  log  cos  <£      

}£  Elapsed  Time  in  arc  =  t.  a.  c.  log  cos  6       

a.  c.  log  sin  t       

Middle  Chronometer  Time.  Log  Correction   

Correction  for  Refraction.  Correction  

Equation  of  Equal  Altitudes.  T" "7 

Chronom.  Time  of  App.  Noon.  u   k 

App.  of  Time  at  App.  Noon.  ....12^.  ,.0m..  .0s. ..      "   tan  <J> 

Eq.  of  Time  at  App.  Noon.  "    1st  Part 

Mean  Time  of  App.  Noon.  .... 1st  Part. 

Error  of  Chronometer  at  App.  Noon Log  B. 

"    k. 

"    tan  8. 

"   2d  Part. 
2d  Part. 
1st  Part  -f-  2d  Part  =  Eq.  of  Equal  Altitudes. 


FORMS. 
JFOBM  No.  7. 


211 


LATITUDE  BY  CIRCUM-MERIDIAN  ALTITUDES  OF 
SUN'S..  ..LIMB. 


Date Station,  WEST  POINT,  N.  Y.    Longitude  4.93ft.    Assumed  Lat.  =  <f>= . 

Observer Recorder Barom Att.  Th Ext.  Th. 

Sextant  No By M.  S.  Chronometer,  No  By 

Error  of  Chronometer  =  E  = Rate  of  Chronometer  =  r  = , 


Observed  Double 
Altitudes. 

0               1               ft 

Chronometer  Times. 
h          m          s 

App.  Time  of 
App.  Noon   12    0    0 
Eq.  of  Time  at 
App.  Noon       

Hour  Angles. 

Wl.               S. 

m. 
s. 

n. 

I. 
II. 
III. 

IV. 
V. 
VI. 
VII. 
VIII. 
IX. 
X. 

Mean  Time  of 
App.  Noon     
Chron.  Error    

Chron.  Time  of 
App.  No<m      

Sum.                    Log.  Eq.  Hor.  Pa.  Sums 

Mean                    "     p                      

P. 

«o 

n0 

Eccentricity       **     cos  a*            Means. 

Index  Error.       ,.     ,                      .     "    Par  in  Alt.        ....... 

Cor.  D.  Alt.                                              "    "    "                 .  .  Eq.  of  Time  at  Pn  .  .  . 

Refraction 
Semi-diam. 
Par.  in  Alt. 
True  Alt.  -a0-|- 


Longitude, 
Correspond'g  Greenwich  Time  . 


Sun's  Dec.  at 


a,  -  S0  +  90°  - 


=  80  +  90°  -  a, 


Sum  — 
90°-J- 


.90°..<y.O..Q".0. 


Change  in  Eq.  of  Time  in  24fe 
Rate  of  Chronometer       " 

i 
log 


A0m0.... 


21ogA0        . 
*'   tan  a,  . 


B0n0 


*  a  is  obtained  by  applying  refraction  and  semi-diameter  to  Corrected  Single  Altitude. 
NOTE.— For  correction  to  Semi-diameter  due  to  difference  of  refraction  between  limb 
and  center,  see  Longitude  by  Lunar  Distances. 


212 


FORMS. 

FORM  No.  8. 


LATITUDE  BY   CIRCUM-MERIDIAN   ALTITUDES   OF  (NAME    OF 
STAR) 

Date Station,  WEST  POINT,  N.  Y.       .  Assumed  Lat.  =0  = 

Observer Recorder Barom. . ,;. Att.  Ther Ext.  Ther 

Sextant  No By Sidereal  Chronom.  No . . . By 

Error  of  Chronometer  =  E  — Rate  of  Chronometer  —  r  = 


Observed  Double 
Altitudes. 

0               /               // 

Chronometer  Times. 
h.         m.         s. 

Hour  Angles. 
m.       s. 

m. 
s. 

n. 
s. 

II. 

App.  R.  A.  of 
Star             

III. 

Chron.  Error  

IV. 
V. 
VI. 

Chron.Timeof 
Transit        

Vlt. 
VII!. 
IX. 
X. 

Sums. 

PO 

7)1  0 

n0 

Means 

Tndfix  F.rror. 

Cor.  D.  Alt Star's  App.  Declination  =  S0 

"    g.     "          o/  =  80  +  90°  —  <fr. 

Refraction.         = 

True  Alt.=a0+ Rate  of  Chronometer  in  24&  =  e. . 

^on,0+  log         k. 

B0n0-  "     cos0. 

Sum-  "    cos80. 

90°  +  .90°..0'.0..0".0.  "    sec  a/ 


=  50  +  90°  -  a,. 


A0mc.. 


2  log 


FORMS. 
FORM  No.  9. 

PROGRAMME  FOR  ZENITH  TELESCOPE.    (LATITUDE.) 


213 


Station,  WEST  POINT,  N.  Y. 


Approximate  Latitude Observer. 


No. 

Catalogue  and 
••         No. 

Mag. 

Mean  R.  A. 

Mean  Dec. 

Zenith 
Dist. 

N. 
S. 

Setting. 

I. 

8. 

3. 
4. 

5. 
6. 

7. 
8. 

9. 
10. 

FORM  No.  9a. 

OBSERVATIONS  WITH  ZENITH  TELESCOPE. 


Station,  WEST  POINT,  N.  Y.           Observer. 
Telescope  No By 


Recorder Sheet  No. 

Chronometer  Error 


DATE. 

STAR. 

MICROMETER. 

LEVEL. 

CHRONOM. 

REMARKS 

Catalogue 
and  Cata- 
logue No. 

N. 
S. 

mn  &  m8 

mn  -  ms 

In&l'n 

Is  &  I's 

(ln+l'n) 
fr+l'.) 

Time. 

NOTE.— Form  No.  9  is  for  the  observer's  use. 
Form  9a  is  for  the  recorder's  use. 

The  records  of  the  different  nights  at  a  given  station  are  then  collated,  and  the 
reductions  made  as  per  Form  No.  96,  which  is  for  the  com  outer's  use. 


214 


FORM& 


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FORMS. 


FORM  No.  11. 

LATITUDE  BY  EQUAL  ALTITUDES  OF  TWO  STARS. 


Station,  Wi 
Date  


POINT,  N.  Y.    Observer Recorder. 

Sextant,  No By 


Sid.  Chronom.  No By Error. 


Rate 


Name 

Chron. 
Time 
of 

True 
Time 
of 

App. 

Hour 

Angle 

•r*' 

Hour 
Angle 

App. 
Dec 

«'  -  S 

8'  +  6 

P'-P 

P'  +  P 

of  the  Star. 

Obser- 
vation. 

Obser- 
vation. 

R.A. 

in 

Time. 

in  Arc. 
P&P' 

S&S' 

i! 

^ 

2 

2 

1. 
2. 

1. 
2. 

1. 
2. 

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2 

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.  S'-6 
Log  cot  —  £— 

Log  cot  —  J- 

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M 

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Log  tan  -J- 

f  +  P 

M 

pf+p   „ 

2 

Log  cos 
F+P      „ 

a 

Log          m 

- 

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t 

Mean 

FORMS. 


217 


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218 


FORMS. 


B 


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OVERDUE. 


AU8  29  1932 


DEC    6  1932 


X"    "SO 


21-20»-6,'82 


YC  22099 


387452 


t^uc 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


